Search: a195825 -id:a195825
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A210843
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Level of the n-th plateau of the column k of the square array A195825, when k -> infinity.
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+20
18
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1, 4, 13, 35, 86, 194, 415, 844, 1654, 3133, 5773, 10372, 18240, 31449, 53292, 88873, 146095, 236977, 379746, 601656, 943305, 1464501, 2252961, 3436182, 5198644, 7805248, 11634685, 17224795, 25336141, 37038139, 53828275, 77792869
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OFFSET
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1,2
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COMMENTS
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Also the first (k+1)/2 terms of this sequence are the levels of the (k+1)/2 plateaus of the column k of A195825, whose lengths are k+1, k-1, k-3, k-5,... 2, if k is odd.
Also the first k/2 terms of this sequence are the levels of the k/2 plateaus of the column k of A195825, whose lengths are k+1, k-1, k-3, k-5,... 3, if k is a positive even number.
For the visualization of the plateaus see the graph of the sequences mentioned in crossrefs section (columns k=1..10 of A195825), for example see the graph of A210964.
Also numbers that are repeated in column k of square array A195825, when k -> infinity.
Note that the definition and the comments related to the square array A195825 mentioned above are also valid for the square array A211970, since both arrays contains the same columns, if k >= 1.
Is this the EULER transform of 4, 3, 3, 3, 3, 3, 3...?
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LINKS
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FORMULA
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a(n) ~ exp(sqrt(2*n)*Pi) / (8*Pi*n).
(End)
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EXAMPLE
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Column 1 of A195825 is A000041 which starts: [1, 1], 2, 3, 5, 7, 11... The column contains only one plateau: [1, 1] which has level 1 and length 2. So a(1) = 1.
Column 3 of A195825 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10... The column contains only two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2. So a(1)= 1 and a(2) = 2.
Column 6 of A195825 is A195850 which starts: [1, 1, 1, 1, 1, 1, 1], 2, 3, [4, 4, 4, 4, 4], 5, 7, 10, 12, [13, 13, 13], 14, 16, 21... The column contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13], which have levels 1, 4, 13 and lengths 7, 5, 3. So a(1) = 1, a(2) = 4 and a(3) = 13.
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MATHEMATICA
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CoefficientList[Series[1/(1-x)*Product[1/(1-x^k)^3, {k, 1, 50}], {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 16 2015 *)
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PROG
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(GW-BASIC)
10 'This program gives the 32 terms of DATA section.
20 'Suppose that we have A057077().
30 'In this case g(n) is the n-th generalized 64-gonal number.
40 DEFDBL a, g, w
50 DIM a(32), A057077(2079), g(2080), w(2079)
60 n=0: w(0)=1
70 FOR i = 1 TO 2079
80 FOR j = 1 TO i
90 IF g(j)<=i THEN w(i)=w(i)+A057077(j-1)*w(i-g(j))
100 NEXT j
110 IF i=1 GOTO 130
120 IF w(i-2)=w(i-1) AND w(i-1)<>a(n) THEN n=n+1: a(n)=w(i-1): PRINT a(n);
130 NEXT i
140 END
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 14, 16, 21, 27, 32, 34, 36, 38, 44, 54, 67, 77, 84, 88, 95, 107, 128, 152, 174, 188, 200, 215, 242, 281, 329, 370, 402, 428, 462, 513, 589, 674, 754, 816, 873, 940, 1041, 1176, 1333, 1477, 1600, 1710, 1845
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OFFSET
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0,7
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COMMENTS
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Note that this sequence contains three plateaus: [1, 1, 1, 1, 1, 1], [4, 4, 4, 4], [13, 13]. For more information see A210843. See also other columns of A195825. - Omar E. Pol, Jun 29 2012
Number of partitions of n into parts congruent to 0, 1 or 6 (mod 7). - Ludovic Schwob, Aug 05 2021
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LINKS
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FORMULA
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G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-1))*(1 - x^(7*k-6))). - Ilya Gutkovskiy, Aug 13 2017
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MAPLE
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7*n^2/8+7*n/8-3/16+3*(-1)^n*(1/16+n/8) ;
end proc:
option remember;
local ks, a, j ;
0 ;
elif n <= 5 then
return 1;
elif k = 1 then
a := 0 ;
for j from 1 do
a := a+procname(n-1, j) ;
else
break;
end if;
end do;
return a;
else
(-1)^floor((k-1)/2)*procname(n-ks+1, 1) ;
end if;
end proc:
end proc:
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MATHEMATICA
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m = 61;
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PROG
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(GW-BASIC)' A program with two A-numbers by Omar E. Pol, Jun 10 2012
20 For n = 1 to 61: For j = 1 to n
40 Next j: Print a(n-1); : Next n
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CROSSREFS
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Cf. A000041, A001082, A006950, A036820, A057077, A118277, A195825, A195829, A195839, A195848, A195850, A195851, A195852, A196933, A210843, A210964, A211971.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 86, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 193, 194, 195
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OFFSET
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0,12
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COMMENTS
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Note that this sequence contains five plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13, 13, 13], [35, 35, 35, 35, 35], [86, 86, 86]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012
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LINKS
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FORMULA
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Expansion of 1 / f(-x, -x^11) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jan 10 2015
Partitions of n into parts of the form 12*k, 12*k+1, 12*k+11. - Michael Somos, Jan 10 2015
Euler transform of period 12 sequence [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, ...]. - Michael Somos, Jan 10 2015
G.f.: Product_{k>0} 1 / ((1 - x^(12*k)) * (1 - x^(12*k - 1)) * (1 - x^(12*k - 11))).
a(n) ~ sqrt(2)*(1+sqrt(3)) * exp(Pi*sqrt(n/6)) / (8*n). - Vaclav Kotesovec, Nov 08 2015
a(n) = a(n-1) + a(n-11) - a(n-14) - a(n-34) + + - - (with the convention a(n) = 0 for negative n), where 1, 11, 14, 34, ... is the sequence of generalized 14-gonal numbers A195818. - Peter Bala, Dec 10 2020
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MATHEMATICA
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nmax = 100; CoefficientList[Series[Product[1 / ((1 - x^(12*k)) * (1 - x^(12*k-1)) * (1 - x^(12*k-11))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)
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PROG
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(GW-BASIC)' A program with two A-numbers:
20 For n = 1 to 67: For j = 1 to n
40 Next j: Print a(n-1); : Next n
50 End
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CROSSREFS
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Cf. A000041, A006950, A036820, A057077, A195818, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A210944, A210954, A211970, A211971.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 36, 38, 44, 54, 67, 77, 83, 86, 89, 95, 107, 128, 152, 173, 186, 194, 202, 216, 242, 281, 328, 368, 396, 415, 434, 464, 514, 588, 672, 748, 803, 844
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graph;
refs;
listen;
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OFFSET
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0,8
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COMMENTS
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Note that this sequence contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012
Number of partitions of n into parts congruent to 0, 1 or 7 (mod 8). - Peter Bala, Dec 10 2020
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LINKS
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FORMULA
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G.f.: Product_{k>=1} 1/((1 - x^(8*k))*(1 - x^(8*k-1))*(1 - x^(8*k-7))). - Ilya Gutkovskiy, Aug 13 2017
a(n) ~ exp(Pi*sqrt(n)/2) / (4*sqrt(2-sqrt(2))*n). - Vaclav Kotesovec, Aug 14 2017
a(n) = a(n-1) + a(n-7) - a(n-10) - a(n-22) + + - - (with the convention a(n) = 0 for negative n), where 1, 7, 10, 22, ... is the sequence of generalized 10-gonal numbers A074377. - Peter Bala, Dec 10 2020
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PROG
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(GW-BASIC)' A program with two A-numbers:
20 For n = 1 to 60: For j = 1 to n
40 Next j: Print a(n-1); : Next n
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CROSSREFS
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Cf. A000041, A001082, A006950, A036820, A057077, A074377, A195825, A195830, A195848, A195849, A195851, A195852, A196933, A210964, A211971.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 87, 89, 95, 107, 128, 152, 173, 185, 192, 196, 203, 216, 242, 281, 328, 367, 394, 409, 421, 436, 465
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,9
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COMMENTS
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Note that this sequence contains four plateaus: [1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4], [13, 13, 13, 13], [35, 35]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012
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LINKS
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FORMULA
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G.f.: Product_{k>=1} 1/((1 - x^(9*k))*(1 - x^(9*k-1))*(1 - x^(9*k-8))). - Ilya Gutkovskiy, Aug 13 2017
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MAPLE
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(18*n*(n+1)+5*(2*n+1)*(-1)^n-5)/16 ;
end proc:
option remember;
local ks, a, j ;
0 ;
elif n <= 5 then
return 1;
elif k = 1 then
a := 0 ;
for j from 1 do
a := a+procname(n-1, j) ;
else
break;
end if;
end do;
return a;
else
(-1)^floor((k-1)/2)*procname(n-ks+1, 1) ;
end if;
end proc:
end proc:
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PROG
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(GW-BASIC)' A program with two A-numbers:
20 For n = 1 to 61: For j = 1 to n
40 Next j: Print a(n-1); : Next n
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CROSSREFS
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Cf. A000041, A001082, A006950, A036820, A057077, A195160, A195831, A195825, A195848, A195849, A195850, A195852, A196933, A210964, A211971.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 193, 195, 197, 203, 216, 242, 281
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,11
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COMMENTS
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Note that this sequence contains five plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13, 13], [35, 35, 35, 35], [86, 86]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012
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LINKS
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FORMULA
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G.f.: Product_{k>=1} 1/((1 - x^(11*k))*(1 - x^(11*k-1))*(1 - x^(11*k-10))). - Ilya Gutkovskiy, Aug 13 2017
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MATHEMATICA
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T := Product[1/((1 - x^(11*k))*(1 - x^(11*k - 1))*(1 - x^(11*k - 10))), {k, 1, 70}]; a:= CoefficientList[Series[T, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 28 2018 *)
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PROG
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(GW-BASIC) ' A program with two A-numbers:
20 For n = 1 to 66: For j = 1 to n
40 Next j: Print a(n-1); : Next n
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CROSSREFS
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Cf. A000041, A001082, A006950, A036820, A057077, A195313, A195825, A195833, A195848, A195849, A195850, A195851, A195852, A210964, A211971.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 194, 197, 203, 216, 242, 281, 328, 367, 393, 407
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,10
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COMMENTS
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Note that this sequence contains four plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13], [35, 35, 35]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012
Number of partitions of n into parts congruent to 0, 1 or 9 (mod 10). - Peter Bala, Dec 10 2020
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LINKS
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FORMULA
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G.f.: Product_{k>=1} 1/((1 - x^(10*k))*(1 - x^(10*k-1))*(1 - x^(10*k-9))). - Ilya Gutkovskiy, Aug 13 2017
a(n) = a(n-1) + a(n-9) - a(n-12) - a(n-28) + + - - (with the convention a(n) = 0 for negative n), where 1, 9, 12, 28, ... is the sequence of generalized 12-gonal numbers A195162. - Peter Bala, Dec 10 2020
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PROG
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(GW-BASIC)' A program with two A-numbers:
20 For n = 1 to 64: For j = 1 to n
40 Next j: Print a(n-1); : Next n
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CROSSREFS
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Cf. A000041, A001082, A006950, A036820, A057077, A195162, A195825, A195832, A195848, A195849, A195850, A195851, A196933, A210964, A211971.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A006950
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G.f.: Product_{k>=1} (1 + x^(2*k - 1)) / (1 - x^(2*k)).
(Formerly M0524)
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+10
67
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1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 130, 161, 196, 236, 287, 350, 420, 501, 602, 722, 858, 1016, 1206, 1431, 1687, 1981, 2331, 2741, 3206, 3740, 4368, 5096, 5922, 6868, 7967, 9233, 10670, 12306, 14193, 16357, 18803, 21581
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OFFSET
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0,4
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COMMENTS
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Also the number of partitions of n in which all odd parts are distinct. There is no restriction on the even parts. E.g., a(9)=13 because "9 = 8+1 = 7+2 = 6+3 = 6+2+1 = 5+4 = 5+3+1 = 5+2+2 = 4+4+1 = 4+3+2 = 4+2+2+1 = 3+2+2+2 = 2+2+2+2+1". - Noureddine Chair, Feb 03 2005
Number of partitions of n in which each even part occurs with even multiplicity. There is no restriction on the odd parts.
Also the number of partitions of n into parts not congruent to 2 mod 4. - James A. Sellers, Feb 08 2002
Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras o(n) of skew-symmetric n X n matrices, n=0,1,2,3,... (the cases n=0,1 being degenerate). This sequence, A015128 and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
Poincaré series [or Poincare series] (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) ⊗ E(e_1, e_2, ... e_n) with b_i 2-dimensional, e_i one-dimensional and the permutation action of S_n, in the case n=2.
It appears that this sequence is related to the generalized hexagonal numbers (A000217) in the same way as the partition numbers A000041 are related to the generalized pentagonal numbers A001318. (See the table in comments section of A195825.) Conjecture: this is 1 together with the row sums of triangle A195836, also column 1 of A195836, also column 2 of the square array A195825. - Omar E. Pol, Oct 09 2011
Since this is also column 2 of A195825 so the sequence contains only one plateau [1, 1, 1] of level 1 and length 3. For more information see A210843. - Omar E. Pol, Jun 27 2012
Also the number of ways to stack n triangles in a valley (pointing upwards or downwards depending on row parity). - Seiichi Manyama, Jul 07 2018
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REFERENCES
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A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 108.
M. D. Hirschhorn, The Power of q, Springer, 2017. See pod, page 297.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Expansion of 1 / psi(-x) in powers of x where psi() is a Ramanujan theta function.
Expansion of q^(1/8) * eta(q^2) / (eta(q) * eta(q^4)) in powers of q.
G.f.: exp( Sum_{n>=1} [Sum_{d|n} (-1)^(n-d)*d] * x^n/n ). - Paul D. Hanna, Jul 22 2009
a(n) ~ (8*n+1) * cosh(sqrt(8*n-1)*Pi/4) / (16*sqrt(2)*n^2) - sinh(sqrt(8*n-1)*Pi/4) / (2*Pi*n^(3/2)) ~ exp(Pi*sqrt(n/2))/(4*sqrt(2)*n) * (1 - (2/Pi + Pi/16)/sqrt(2*n) + (3/16 + Pi^2/1024)/n). - Vaclav Kotesovec, Aug 17 2015, extended Jan 09 2017
Can be computed recursively by Sum_{j>=0} (-1)^(ceiling(j/2)) a(n - j(j+1)/2) = 0, for n > 0. [Merca, Theorem 4.3] - Eric M. Schmidt, Sep 21 2017
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EXAMPLE
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G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 13*x^9 + ...
G.f. = q^-1 + q^7 + q^15 + 2*q^23 + 3*q^31 + 4*q^39 + 5*q^47 + 7*q^55 + 10*q^63 + ...
n | the ways to stack n triangles in a valley
--+------------------------------------------------------
1 | *---*
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| *
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2 | *
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| *
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3 | *---* *---*
| / \ / \ / \
| *---* *---*
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| * *
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4 | * *
| / \ / \
| *---* *---*---* *---*
| / \ / \ / \ / \ / \
| *---* *---* *---*
| \ / \ / \ /
| * * *
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5 | *---* * * *---*
| / \ / / \ / \ \ / \
| *---* *---*---* *---*---* *---*
| / \ / \ / \ / \ / \ / \ / \
| *---* *---* *---* *---*
| \ / \ / \ / \ /
| * * * *
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6 | *
| / \
| *---* *---* * * *---*
| / \ / / \ / / \ / \ \ / \
| *---* *---*---* *---*---* *---*---*
| / \ / \ / \ / \ / \ / \ / \ /
| *---* *---* *---* *---*
| \ / \ / \ / \ /
| * * * *
| *
| / \
| *---*
| \ / \
| *---*
| \ / \
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| \ /
| *
(End)
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+`if`(i>n, 0, b(n-i, i-irem(i, 2)))))
end:
a:= n-> b(n, n):
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MATHEMATICA
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CoefficientList[ Series[ Product[(1 + x^(2k - 1))/(1 - x^(2k)), {k, 25}], {x, 0, 50}], x] (* Robert G. Wilson v, Jun 28 2012 *)
CoefficientList[Series[x*QPochhammer[-1/x, x^2] / ((1+x)*QPochhammer[x^2, x^2]), {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
CoefficientList[Series[2*(-x)^(1/8) / EllipticTheta[2, 0, Sqrt[-x]], {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i-Mod[i, 2]]]]];
a[n_] := b[n, n];
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PROG
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(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, (-1)^(m-d)*d)*x^m/m)+x*O(x^n)), n)} \\ Paul D. Hanna, Jul 22 2009
(GW-BASIC)' A program with two A-numbers (Note that here A000217 are the generalized hexagonal numbers):
20 For n = 1 to 51: For j = 1 to n
40 Next j: Print a(n-1); : Next n ' Omar E. Pol, Jun 10 2012
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A036820
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Number of partitions satisfying (cn(2,5) = cn(3,5) = 0).
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+10
21
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1, 1, 1, 1, 2, 3, 4, 4, 5, 7, 10, 12, 14, 16, 21, 27, 33, 37, 44, 54, 68, 80, 92, 106, 129, 155, 182, 207, 240, 283, 337, 389, 444, 508, 594, 692, 797, 902, 1030, 1187, 1373, 1564, 1770, 2004, 2295, 2624, 2978, 3349, 3783, 4293, 4880, 5501, 6174, 6932, 7830, 8834
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OFFSET
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0,5
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COMMENTS
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For a given partition cn(i,n) means the number of its parts equal to i modulo n.
Short: (2=3 := 0).
It appears that this sequence is related to the generalized heptagonal numbers A085787 in the same way as the partition numbers A000041 are related to the generalized pentagonal numbers A001318. (See the table in comments section of A195825.) Conjecture: Column 1 of triangle A195837. Also 1 together with the row sums of triangle A195837. Also column 3 of the square array A195825. - Omar E. Pol, Oct 08 2011
Note that this sequence contains two plateaus: [1, 1, 1, 1] and [4, 4]. For more information see A195825 and A210843. - Omar E. Pol, Jun 23 2012
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LINKS
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FORMULA
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Euler transform of period 5 sequence [1, 0, 0, 1, 1, ...]. - Michael Somos, Feb 09 2012
Expansion of 1 / f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 08 2012
G.f.: 1 / (Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 1)) * (1 - x^(5*k - 4))). - Michael Somos, Sep 08 2012
G.f.: 1 / (Sum_{k in Z} (-1)^k * x^(k * (5*k + 3) / 2)). - Michael Somos, Sep 08 2012
a(n) ~ sqrt(1+sqrt(5)) * exp(sqrt(2*n/5)*Pi) / (2^(5/2)*5^(1/4)*n). - Vaclav Kotesovec, Oct 06 2015
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EXAMPLE
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G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 10*x^10 + ...
G.f. = q^-9 + q^31 + q^71 + q^111 + 2*q^151 + 3*q^191 + 4*q^231 + 4*q^271 + 5*q^311 + ... - Michael Somos, Sep 08 2012
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MAPLE
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with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*[1, 1, 0, 0, 1]
[1+irem(d, 5)], d=divisors(j))*a(n-j), j=1..n)/n)
end:
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MATHEMATICA
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a[n_] := a[n] = If[n == 0, 1, Sum[ Sum[ d*{1, 1, 0, 0, 1}[[1 + Mod[d, 5]]], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 15 2015, after Alois P. Heinz *)
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PROG
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(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, (n+4)\5, (1 - x^(5*k - 4)) * (1 - x^(5*k - 1)) * (1 - x^(5*k)), 1 + x * O(x^n)), n))}; /* Michael Somos, Feb 09 2012 */
(GW-BASIC)' A program with two A-numbers:
20 For n = 1 to 56: For j = 1 to n
40 Next j: Print a(n-1); : Next n ' Omar E. Pol, Jun 10 2012
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A195848
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Expansion of 1 / f(-x^1, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.
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+10
19
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1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 5, 7, 10, 12, 13, 14, 16, 21, 27, 32, 35, 38, 44, 54, 67, 78, 86, 94, 107, 128, 153, 176, 194, 213, 241, 282, 331, 376, 415, 456, 512, 590, 680, 767, 845, 928, 1037, 1180, 1345, 1506, 1657, 1818, 2020, 2278, 2570, 2862, 3142, 3442
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OFFSET
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0,6
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COMMENTS
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Also column 4 of A195825, therefore this sequence contains two plateaus: [1, 1, 1, 1, 1], [4, 4, 4]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 26 2012
The number of partitions of n into parts congruent to 0, 1 or 5 ( mod 6 ). - Peter Bala, Dec 09 2020
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LINKS
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FORMULA
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Expansion of 1 / (psi(x^3) * chi(-x)) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Jun 07 2012
Expansion of q^(1/3) * eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)^2) in powers of q. - Michael Somos, Jun 07 2012
Euler transform of period 6 sequence [ 1, 0, 0, 0, 1, 1, ...]. - Michael Somos, Oct 18 2014
O.g.f.: 1/( Product_{n >= 1} (1 - x^(6*n-5))*(1 - x^(6*n-1))*(1 - x^(6*n)) ).
a(n) = a(n-1) + a(n-5) - a(n-8) - a(n-16) + + - - ... (with the convention a(n) = 0 for negative n), where 1, 5, 8, 16, ... is the sequence of generalized octagonal numbers A001082. (End)
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EXAMPLE
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G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 4*x^8 + 4*x^9 + 5*x^10 + ...
G.f. = 1/q + q^2 + q^5 + q^8 + q^11 + 2*q^14 + 3*q^17 + 4*q^20 + 4*q^23 + 4*q^26 + ...
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MAPLE
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if type(n, 'even') then
n*(3*n-4)/4 ;
else
(n-1)*(3*n+1)/4 ;
end if;
end proc:
option remember;
local ks, a, j ;
0 ;
elif n <= 5 then
return 1;
elif k = 1 then
a := 0 ;
for j from 1 do
a := a+procname(n-1, j) ;
else
break;
end if;
end do;
return a;
else
(-1)^floor((k-1)/2)*procname(n-ks+1, 1) ;
end if;
end proc:
end proc:
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]^2), {x, 0, n}]; (* Michael Somos, Oct 18 2014 *)
a[ n_] := SeriesCoefficient[ 2 q^(3/8) / (QPochhammer[ q, q^2] EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}]; (* Michael Somos, Oct 18 2014 *)
nmax = 60; CoefficientList[Series[Product[(1+x^k) / ((1+x^(3*k)) * (1-x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)
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PROG
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(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)^2), n))}; /* Michael Somos, Jun 07 2012 */
(GW-BASIC)' A program with two A-numbers:
20 For n = 1 to 58: For j = 1 to n
40 Next j: Print a(n-1); : Next n (End)
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CROSSREFS
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Column 1 of triangle A195838. Also 1 together with the row sums of triangle A195838. Column 4 of array A195825.
Cf. A000041, A001082, A006950, A036820, A057077, A195825, A195828, A195849, A195850, A195851, A195852, A196933, A210843, A210964, A211971.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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