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Search: a211970 -id:a211970
Displaying 1-10 of 12 results found. page 1 2
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A211971 Column 0 of square array A211970 (in which column 1 is A000041). +20
19
1, 1, 2, 4, 6, 10, 16, 24, 36, 54, 78, 112, 160, 224, 312, 432, 590, 802, 1084, 1452, 1936, 2568, 3384, 4440, 5800, 7538, 9758, 12584, 16160, 20680, 26376, 33520, 42468, 53644, 67552, 84832, 106246, 132706, 165344, 205512, 254824, 315256, 389168, 479368 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Partial sums give A015128. - Omar E. Pol, Jan 09 2014
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
FORMULA
a(n) ~ exp(Pi*sqrt(n))*Pi / (16*n^(3/2)) * (1 - (3/Pi + Pi/4)/sqrt(n) + (3/2 + 3/Pi^2+ Pi^2/24)/n). - Vaclav Kotesovec, Oct 25 2016, extended Nov 04 2016
G.f.: (1 - x)/theta_4(x), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Mar 05 2018
MATHEMATICA
Flatten[{1, Differences[Table[Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 0, 60}]]}] (* Vaclav Kotesovec, Oct 25 2016 *)
CoefficientList[Series[(1 - x)/EllipticTheta[4, 0, x], {x, 0, 43}], x] (* Robert G. Wilson v, Mar 06 2018 *)
PROG
(GW-BASIC)' A program with two A-numbers:
10 Dim A008794(100), A057077(100), a(100): a(0)=1
20 For n = 1 to 43: For j = 1 to n
30 If A008794(j+1) <= n then a(n) = a(n) + A057077(j-1)*a(n - A008794(j+1))
40 Next j: Print a(n-1); : Next n
CROSSREFS
Cf. A000041, A006950, A008794, A036820, A057077, A195152, A195848, A195849, A195850, A195851, A195852, A196933, A195825, A210964, A277643.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jun 10 2012
STATUS
approved
A235670 Square array read by antidiagonals upwards in which the n-th column gives the partial sums of the n-th column of A211970. +20
0
1, 2, 1, 4, 2, 1, 8, 4, 2, 1, 14, 7, 3, 2, 1, 24, 12, 5, 3, 2, 1, 40, 19, 8, 4, 3, 2, 1, 64, 30, 12, 6, 4, 3, 2, 1, 100, 45, 17, 9, 5, 4, 3, 2, 1, 154, 67, 24, 13, 7, 5, 4, 3, 2, 1, 232, 97, 34, 17, 10, 8, 6, 5, 4, 3, 2, 1, 344, 139, 47, 22, 14, 8, 6, 5, 4, 3, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The column 0 is related to A008794 in the same way as the column k is related to the generalized (k+4)-gonal numbers, for k >= 1. For more information see A195152 and A211970.
LINKS
Table of n, a(n) for n=0..78.
FORMULA
T(n,k) = Sum_{j=0..n} A211970(j,k), (n>=0, k>=0).
EXAMPLE
Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,...
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,...
4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3,...
8, 7, 5, 4, 4, 4, 4, 4, 4, 4, 4,...
14, 12, 8, 6, 5, 5, 5, 5, 5, 5, 5,...
24, 19, 12, 9, 7, 6, 6, 6, 6, 6, 6,...
40, 30, 17, 13, 10, 8, 7, 7, 7, 7, 7,...
64, 45, 24, 17, 14, 11, 9, 8, 8, 8, 8,...
100, 67, 34, 22, 18, 15, 12, 10, 9, 9, 9,...
154, 97, 47, 29, 22, 19, 16, 13, 11, 10, 10,...
232, 139, 63, 39, 27, 23, 20, 17, 14, 12, 11,...
344, 195, 84, 51, 34, 27, 24, 21, 18, 15, 13,...
504, 272, 112, 65, 44, 32, 28, 25, 22, 19, 16,...
728, 383, 147, 81, 56, 39, 32, 29, 26, 23, 20,...
...
CROSSREFS
Column 1 is A015128, the partial sums of A211971.
Column 2 is A000070, the partial sums of A000041.
Column 3 is A233969, the partial sums of A006950.
Cf. A000217, A001318, A008794, A085787, A057077, A195152, A211970.
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Jan 13 2014
STATUS
approved
A195825 Square array T(n,k) read by antidiagonals, n>=0, k>=1, which arises from a generalization of Euler's Pentagonal Number Theorem. +10
38
1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 7, 3, 1, 1, 1, 1, 11, 4, 2, 1, 1, 1, 1, 15, 5, 3, 1, 1, 1, 1, 1, 22, 7, 4, 2, 1, 1, 1, 1, 1, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1, 56, 16, 7, 4, 3, 1, 1, 1, 1, 1, 1, 1, 77, 21, 10, 4 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
In the infinite square array the column k is related to the generalized m-gonal numbers, where m = k+4. For example: the first column is related to the generalized pentagonal numbers A001318. The second column is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on ... (see the program in which A195152 is a table of generalized m-gonal numbers).
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041 (see below the first row of the table):
========================================================
. Column k of
. this square
. Generalized Triangle Triangle array A195825
k m m-gonal "A" "B" [row sums of
. numbers triangle "B"
. with a(0)=1]
========================================================
1 5 A001318 A195310 A175003 A000041
2 6 A000217 A195826 A195836 A006950
3 7 A085787 A195827 A195837 A036820
4 8 A001082 A195828 A195838 A195848
5 9 A118277 A195829 A195839 A195849
6 10 A074377 A195830 A195840 A195850
7 11 A195160 A195831 A195841 A195851
8 12 A195162 A195832 A195842 A195852
9 13 A195313 A195833 A195843 A196933
10 14 A195818 A210944 A210954 A210964
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
Conjecture: if k is odd then column k contains (k+1)/2 plateaus whose levels are the first (k+1)/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 2. Otherwise, if k is even then column k contains k/2 plateaus whose levels are the first k/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 3. The sequence A210843 gives the levels of the plateaus of column k, when k -> infinity. For the visualization of the plateaus see the graph of a column, for example see the graph of A210964. - Omar E. Pol, Jun 21 2012
LINKS
Table of n, a(n) for n=0..81.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem
FORMULA
Column k is asymptotic to exp(Pi*sqrt(2*n/(k+2))) / (8*sin(Pi/(k+2))*n). - Vaclav Kotesovec, Aug 14 2017
EXAMPLE
Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
3, 2, 1, 1, 1, 1, 1, 1, 1, 1, ...
5, 3, 2, 1, 1, 1, 1, 1, 1, 1, ...
7, 4, 3, 2, 1, 1, 1, 1, 1, 1, ...
11, 5, 4, 3, 2, 1, 1, 1, 1, 1, ...
15, 7, 4, 4, 3, 2, 1, 1, 1, 1, ...
22, 10, 5, 4, 4, 3, 2, 1, 1, 1, ...
30, 13, 7, 4, 4, 4, 3, 2, 1, 1, ...
42, 16, 10, 5, 4, 4, 4, 3, 2, 1, ...
56, 21, 12, 7, 4, 4, 4, 4, 3, 2, ...
77, 28, 14, 10, 5, 4, 4, 4, 4, 3, ...
101, 35, 16, 12, 7, 4, 4, 4, 4, 4, ...
135, 43, 21, 13, 10, 5, 4, 4, 4, 4, ...
176, 55, 27, 14, 12, 7, 4, 4, 4, 4, ...
...
Column 1 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.
Column 3 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10, ... The column contains two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2.
Column 6 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.
PROG
(GW-BASIC)' A program (with two A-numbers) for the table of example section.
10 DIM A057077(100), A195152(15, 10), T(15, 10)
20 FOR k = 1 TO 10 'Column 1-10
30 T(0, k) = 1 'Row 0
40 FOR n = 1 TO 15 'Rows 1-15
50 FOR j = 1 TO n
60 IF A195152(j, k) <= n THEN T(n, k) = T(n, k) + A057077(j-1) * T(n - A195152(j, k), k)
70 NEXT j
80 NEXT n
90 NEXT k
100 FOR n = 0 TO 15
110 FOR j = 1 TO 10
120 PRINT T(n, k);
130 NEXT k
140 PRINT
150 NEXT n
160 END
170 'Omar E. Pol, Jun 18 2012
CROSSREFS
Columns (1-10): A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
For another version see A211970.
Cf. A057077, A195152, A210843.
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Sep 24 2011
STATUS
approved
A210843 Level of the n-th plateau of the column k of the square array A195825, when k -> infinity. +10
18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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...?
LINKS
Table of n, a(n) for n=1..32.
FORMULA
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ sqrt(2*n)/Pi * A000716(n).
a(n) ~ exp(sqrt(2*n)*Pi) / (8*Pi*n).
(End)
EXAMPLE
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.
MATHEMATICA
CoefficientList[Series[1/(1-x)*Product[1/(1-x^k)^3, {k, 1, 50}], {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 16 2015 *)
PROG
(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
CROSSREFS
Partial sums of A000716. Column 3 of A210764.
Columns (k=1..10) of A195825: A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
Cf. A000070, A000712, A000713, A010815, A211970.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jun 19 2012
STATUS
approved
A210964 Column 10 of square array A195825. Also column 1 of triangle A210954. Also 1 together with the row sums of triangle A210954. +10
16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,12
COMMENTS
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
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..3000 from Vaclav Kotesovec)
Vaclav Kotesovec, Graph - The asymptotic ratio
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
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))).
Convolution inverse of A247133.
a(n) ~ sqrt(2)*(1+sqrt(3)) * exp(Pi*sqrt(n/6)) / (8*n). - Vaclav Kotesovec, Nov 08 2015
a(n) = (1/n)*Sum_{k=1..n} A284372(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
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
MATHEMATICA
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 *)
PROG
(GW-BASIC)' A program with two A-numbers:
10 Dim A195818(100), A057077(100), a(100): a(0)=1
20 For n = 1 to 67: For j = 1 to n
30 If A195818(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A195818(j))
40 Next j: Print a(n-1); : Next n
50 End
CROSSREFS
Cf. A000041, A006950, A036820, A057077, A195818, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A210944, A210954, A211970, A211971.
Cf. A247133.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jun 16 2012
STATUS
approved
A210764 Square array T(n,k), n>=0, k>=0, read by antidiagonals in which column k gives the partial sums of column k of A144064. +10
3
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 8, 4, 1, 1, 12, 18, 13, 5, 1, 1, 19, 38, 35, 19, 6, 1, 1, 30, 74, 86, 59, 26, 7, 1, 1, 45, 139, 194, 164, 91, 34, 8, 1, 1, 67, 249, 415, 416, 281, 132, 43, 9, 1, 1, 97, 434, 844, 990, 787, 447, 183, 53, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
It appears that row 2 is A034856.
Observation:
Column 1 is the EULER transform of 2,1,1,1,1,1,1,1...
Column 2 is the EULER transform of 3,2,2,2,2,2,2,2...
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
EXAMPLE
Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
1, 4, 8, 13, 19, 26, 34, 43, 53,
1, 7, 18, 35, 59, 91, 132, 183,
1, 12, 38, 86, 164, 281, 447,
1, 19, 74, 194, 416, 787,
1, 30, 139, 415, 990,
1, 45, 249, 844,
1, 67, 434,
1, 97,
1,
MAPLE
with(numtheory):
etr:= proc(p) local b;
b:= proc(n) option remember; `if`(n=0, 1,
add(add(d*p(d), d=divisors(j))*b(n-j), j=1..n)/n)
end
end:
A:= (n, k)-> etr(j-> k +`if`(j=1, 1, 0))(n):
seq(seq(A(d-k, k), k=0..d), d=0..14); # Alois P. Heinz, May 20 2013
MATHEMATICA
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[Function[{j}, k + If[j == 1, 1, 0]]][n]; Table[Table[A[d-k, k], {k, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)
CROSSREFS
Columns (0-3): A000012, A000070, A000713, A210843.
Rows (0-1): A000012, A000027.
Main diagonal gives A303070.
Cf. A000007, A000041, A005758, A006922, A000712, A000716, A023003-A023021, A144064, A195825, A211970.
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Jun 27 2012
STATUS
approved
A233758 Bisection of A006950 (the even part). +10
3
1, 1, 3, 5, 10, 16, 28, 43, 70, 105, 161, 236, 350, 501, 722, 1016, 1431, 1981, 2741, 3740, 5096, 6868, 9233, 12306, 16357, 21581, 28394, 37128, 48406, 62777, 81182, 104494, 134131, 171467, 218607, 277691, 351841, 444314, 559727, 703002, 880896, 1100775 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
See Zaletel-Mong paper, page 14, FIG. 11: C2a is this sequence, C2b is A233759, C2c is A015128.
LINKS
Table of n, a(n) for n=1..42.
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el] (2012), 14 (C2a).
MATHEMATICA
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[2 n - 2, 2 n - 2];
Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz in A006950 *)
CROSSREFS
Cf. A000009, A015128, A006950, A069910, A069911, A195825, A195836, A211970, A233759.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 11 2014
STATUS
approved
A233759 Bisection of A006950 (the odd part). +10
3
1, 2, 4, 7, 13, 21, 35, 55, 86, 130, 196, 287, 420, 602, 858, 1206, 1687, 2331, 3206, 4368, 5922, 7967, 10670, 14193, 18803, 24766, 32490, 42411, 55159, 71416, 92152, 118434, 151725, 193676, 246491, 312677, 395537, 498852, 627509, 787171, 985043, 1229494 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
See Zaletel-Mong paper, page 14, FIG. 11: C2a is A233758, C2b is this sequence, C2c is A015128.
LINKS
Table of n, a(n) for n=1..42.
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el] (2012), 14 (C2b).
MATHEMATICA
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[2 n - 1, 2 n - 1];
Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz in A006950 *)
CROSSREFS
Cf. A000009, A015128, A006950, A069910, A069911, A195825, A195836, A211970, A233758.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 11 2014
STATUS
approved
A233969 Partial sums of A006950. +10
2
1, 2, 3, 5, 8, 12, 17, 24, 34, 47, 63, 84, 112, 147, 190, 245, 315, 401, 506, 636, 797, 993, 1229, 1516, 1866, 2286, 2787, 3389, 4111, 4969, 5985, 7191, 8622, 10309, 12290, 14621, 17362, 20568, 24308, 28676, 33772, 39694, 46562, 54529, 63762, 74432, 86738 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The first three columns of A211970 are A211971, A000041, A006950, so for k = 0..2, the partial sums of column k of A211970 give: A015128, A000070, this sequence.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
a(n) ~ exp(Pi*sqrt(n/2))/(2*Pi*sqrt(n)). - Vaclav Kotesovec, Oct 27 2016
MAPLE
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:= proc(n) option remember; b(n, n) +`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..50); # Alois P. Heinz, Jan 12 2014
MATHEMATICA
Accumulate[CoefficientList[Series[x*QPochhammer[-1/x, x^2]/((1 + x) * QPochhammer[x^2]), {x, 0, 50}], x]] (* Vaclav Kotesovec, Oct 27 2016 *)
CROSSREFS
Cf. A000041, A000070, A015128, A195825, A195826, A195836, A210843, A211970, A211971, A233758, A233759.
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 12 2014
STATUS
approved
A249120 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers of A210843 multiplied by A000330(k), and the first element of column k is in row A000217(k). +10
2
1, 4, 13, 5, 35, 20, 86, 65, 194, 175, 14, 415, 430, 56, 844, 970, 182, 1654, 2075, 490, 3133, 4220, 1204, 30, 5773, 8270, 2716, 120, 10372, 15665, 5810, 390, 18240, 28865, 11816, 1050, 31449, 51860, 23156, 2580, 53292, 91200, 43862, 5820, 55, 88873, 157245, 80822, 12450, 220, 146095, 266460, 145208, 25320, 715 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Conjecture: gives an identity for the sum of all divisors of all positive integers <= n. Alternating sum of row n equals A024916(n), i.e., sum_{k=1..A003056(n))} (-1)^(k-1)*T(n,k) = A024916(n).
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
Column 1 is A210843.
Column k lists the partial sums of the k-th column of triangle A252117 which gives an identity for sigma.
The first element of column k is A000330(k).
The second element of column k is A002492(k).
LINKS
Table of n, a(n) for n=1..55.
EXAMPLE
Triangle begins:
1;
4;
13, 5;
35, 20;
86, 65;
194, 175, 14;
415, 430, 56;
844, 970, 182;
1654, 2075, 490;
3133, 4220, 1204, 30;
5773, 8270, 2716, 120;
10372, 15665, 5810, 390;
18240, 28865, 11816, 1050;
31449, 51860, 23156, 2580;
53292, 91200, 43862, 5820, 55;
88873, 157245, 80822, 12450, 220;
146095, 266460, 145208, 25320, 715;
236977, 444365, 255360, 49620, 1925;
379746, 730475, 440286, 93990, 4730;
601656, 1184885, 746088, 173190, 10670;
943305, 1898730, 1244222, 311160, 22825, 91;
...
For n = 6 the sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 1 + 3 + 4 + 7 + 6 + 12 = 33. On the other hand the 6th row of triangle is 194, 175, 14, so the alternating row sum is 194 - 175 + 14 = 33, equaling the sum of all divisors of all positive integers <= 6.
CROSSREFS
Cf. A000203, A000217, A000330, A002492, A003056, A024916, A195825, A196020, A210843, A211970, A236104, A252117.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 14 2014
STATUS
approved
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Last modified August 29 21:13 EDT 2024. Contains 375518 sequences. (Running on oeis4.)