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Search: a143064 -id:a143064
Displaying 1-8 of 8 results found. page 1
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A185646 Square array A(n,m), n>=0, m>=0, read by antidiagonals, where column m is the expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))). +10
12
1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, -1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 1, 2, 3, 3, 1, 0, 1, 1, 1, 2, 3, 4, 5, 1, -1, 1, 1, 1, 2, 3, 5, 7, 6, 1, 0, 1, 1, 1, 2, 3, 5, 8, 11, 10, 1, 0, 1, 1, 1, 2, 3, 5, 9, 13, 17, 14, 1, 0, 1, 1, 1, 2, 3, 5, 9, 14, 22, 28, 21, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,19
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013
EXAMPLE
Square array A(n,m) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, 2, ...
0, 1, 2, 3, 3, 3, 3, 3, 3, ...
-1, 1, 3, 4, 5, 5, 5, 5, 5, ...
0, 1, 5, 7, 8, 9, 9, 9, 9, ...
0, 1, 6, 11, 13, 14, 15, 15, 15, ...
-1, 1, 10, 17, 22, 24, 25, 26, 26, ...
MATHEMATICA
nMax = 12; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A = Table[col[m][[1 ;; nMax + 1]], {m, 0, nMax}] // Transpose; a[n_ /; 0 <= n <= nMax, m_ /; 0 <= m <= nMax] := With[{n1 = n + 1, m1 = m + 1}, A[[n1, m1]]]; Table[a[n - m, m], {n, 0, nMax}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2016 *)
CROSSREFS
Columns m=0-10 give: A143064, A000012, A227360, A173173(n+1), A227374, A227375, A228646, A228644, A185648, A228645, A185649.
Diagonal gives: A005169.
KEYWORD
sign,tabl
AUTHOR
Alois P. Heinz, Aug 29 2013
STATUS
approved
A228644 Expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))) for m=7. +10
7
1, 1, 1, 2, 3, 5, 9, 15, 26, 44, 76, 131, 225, 389, 670, 1156, 1994, 3439, 5934, 10236, 17661, 30470, 52569, 90699, 156483, 269985, 465811, 803677, 1386609, 2392357, 4127611, 7121498, 12286951, 21199078, 36575462, 63104849, 108876873, 187848862, 324101847 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013
Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,0,-1,-1,-3,-2,-1,0,2,2,3,3,1,0,0,-2,-1,-1,-1).
FORMULA
G.f.: -(x^18 +x^17 +x^16 +2*x^15 +x^14 -2*x^11 -2*x^10 -2*x^9 -2*x^8 +x^5 +x^4 +x^3 +x^2-1) / ((x-1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)*(x^15 +x^14 +x^13 +2*x^12 -x^9 -2*x^8 -2*x^7 -x^6 +x^3 +x^2 +x-1)).
MAPLE
a:= n-> coeff(series(-(x^18 +x^17 +x^16 +2*x^15 +x^14 -2*x^11 -2*x^10 -2*x^9 -2*x^8 +x^5 +x^4 +x^3 +x^2-1) / ((x-1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)*(x^15 +x^14 +x^13 +2*x^12 -x^9 -2*x^8 -2*x^7 -x^6 +x^3 +x^2 +x-1)), x, n+1), x, n): seq(a(n), n=0..50);
MATHEMATICA
nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A228644 = col[7][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)
CROSSREFS
Cf. A143064 (m=0), A227360 (m=2), A173173 (m=3), A227374 (m=4), A227375 (m=5), A228646(m=6), A228645 (m=9).
Column m=7 of A185646.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 28 2013
STATUS
approved
A228645 Expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))) for m=9. +10
7
1, 1, 1, 2, 3, 5, 9, 15, 26, 45, 78, 134, 232, 402, 695, 1205, 2086, 3613, 6259, 10841, 18780, 32531, 56354, 97621, 169111, 292954, 507488, 879136, 1522947, 2638242, 4570298, 7917253, 13715281, 23759370, 41159039, 71300984, 123516755, 213971647, 370669282 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 0, 0, -1, -1, -2, -1, -3, -2, 0, 1, 3, 4, 4, 4, 4, 2, 0, -2, -3, -5, -4, -4, -3, -2, 0, 1, 1, 2, 2, 1, 1, 1).
FORMULA
G.f.: -(x^30 +x^29 +x^28 +2*x^27 +2*x^26 +2*x^25 +x^24 +x^23 -x^22 -2*x^21 -2*x^20 -4*x^19 -4*x^18 -3*x^17 -2*x^16 -x^15 +2*x^13 +2*x^12 +3*x^11 +3*x^10 +x^9 +x^8 -x^5 -x^4 -x^3 -x^2+1) / ((x-1)*(x^2 +x+1)*(x^6 +x^3+1)*(x^26 +x^25 +x^24 +2*x^23 +2*x^22 +x^21 +x^20 -2*x^18 -2*x^17 -3*x^16 -3*x^15 -3*x^14 -x^13 -x^12 +x^11 +2*x^10 +2*x^9 +2*x^8 +x^7 +x^6 -x^3 -x^2 -x+1)).
MAPLE
a:= n-> coeff(series(-(x^30 +x^29 +x^28 +2*x^27 +2*x^26 +2*x^25 +x^24 +x^23 -x^22 -2*x^21 -2*x^20 -4*x^19 -4*x^18 -3*x^17 -2*x^16 -x^15 +2*x^13 +2*x^12 +3*x^11 +3*x^10 +x^9 +x^8 -x^5 -x^4 -x^3 -x^2+1) / ((x-1)*(x^2 +x+1)*(x^6 +x^3+1)*(x^26 +x^25 +x^24 +2*x^23 +2*x^22 +x^21 +x^20 -2*x^18 -2*x^17 -3*x^16 -3*x^15 -3*x^14 -x^13 -x^12 +x^11 +2*x^10 +2*x^9 +2*x^8 +x^7 +x^6 -x^3 -x^2 -x+1)), x, n+1), x, n): seq(a(n), n=0..50);
MATHEMATICA
nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x] &; A228645 = col[9][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)
CROSSREFS
Cf. A143064 (m=0), A227360 (m=2), A173173 (m=3), A227374 (m=4), A227375 (m=5), A228646 (m=6), A228644 (m=7).
Column m=9 of A185646.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 28 2013
STATUS
approved
A228646 Expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))) for m=6. +10
4
1, 1, 1, 2, 3, 5, 9, 15, 25, 43, 74, 126, 217, 372, 638, 1096, 1881, 3230, 5546, 9524, 16353, 28083, 48224, 82811, 142208, 244204, 419360, 720144, 1236670, 2123670, 3646879, 6262611, 10754485, 18468174, 31714525, 54461873, 93524824, 160605817, 275800867 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013
MATHEMATICA
nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A228646 = col[6][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)
CROSSREFS
Cf. A143064 (m=0), A227360 (m=2), A173173 (m=3), A227374 (m=4), A227375 (m=5), A228644 (m=7), A228645 (m=9).
Column m=6 of A185646.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 28 2013
STATUS
approved
A292420 Expansion of a q-series used by Ramanujan in his Lost Notebook. +10
2
1, 2, 2, 3, 4, 4, 6, 8, 8, 11, 14, 16, 20, 24, 28, 34, 42, 48, 57, 68, 78, 94, 110, 126, 148, 172, 198, 230, 266, 304, 351, 404, 460, 526, 602, 684, 780, 888, 1004, 1140, 1290, 1456, 1646, 1856, 2088, 2351, 2644, 2964, 3326, 3728, 4168, 4664, 5212, 5812, 6484 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=-1.
LINKS
Robert Israel, Table of n, a(n) for n = 0..1000
FORMULA
G.f. is the product of the g.f. of A000009 and A143064.
Given g.f. A(x), then A(x^2) = 1 / (1+x) + x / (1+x^3) + x^2 * (1+x^2) / ((1+x^3) * (1+x^5)) + x^3 * (1+x^2) / ((1+x^5) * (1+x^7)) + x^4 * (1+x^2) * (1+x^4) / ((1+x^5) * (1+x^7) * (1+x^9)) + ...
EXAMPLE
G.f. = 1 + 2*x + 2*x^2 + 3*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + ...
MAPLE
N:= 200: # to get a(0)..a(N)
g143064:= add(x^k/mul(1+x^(2*j+1), j=0..k), k=0..2*N):
g000009:= mul(1+x^(2*k), k=1..N):
S:= series(g143064*g000009, x, 2*N+2):
seq(coeff(S, x, 2*j), j=0..N); # Robert Israel, Sep 17 2017
MATHEMATICA
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ QPochhammer[ x^2] / QPochhammer[ x] Sum[ (-1)^k x^(3 k^2 + 2 k) (1 + x^(2 k + 1)), {k, 0, Sqrt[n / 3]}], {x, 0, n}]];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A) * sum(k=0, sqrtint(n \ 3), (-1)^k * x^(3*k^2 + 2*k) * (1 + x^(2*k + 1)), A), n))};
CROSSREFS
Cf. A000009, A143064, A292445.
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 16 2017
STATUS
approved
A143065 Expansion of quotient of a Ramanujan false theta series by the theta series of triangular numbers in powers of x. +10
1
1, 0, 0, -1, 1, -2, 2, -3, 4, -5, 6, -8, 11, -13, 16, -21, 27, -32, 39, -49, 61, -73, 87, -107, 131, -155, 184, -223, 267, -315, 372, -443, 526, -617, 722, -852, 1002, -1167, 1359, -1590, 1854, -2148, 2488, -2888, 3346, -3859, 4444, -5128, 5909, -6779, 7773 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
REFERENCES
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 13th equation.
LINKS
Table of n, a(n) for n=0..50.
FORMULA
G.f.: ( 1 + x - x^5 - x^8 + x^16 + x^21 - ... ) / ( 1 + x + x^3 + x^6 + x^10 + x^15 + ... ). [Ramanujan]
G.f.: 1 - x^3 * (1 - x) / (1 - x^2)^2 + x^8 * (1 - x) * (1 - x^3) / ((1 - x^2)^2 * (1 - x^4)^2) - ... [Ramanujan]
Convolution with A010054 is A143064.
EXAMPLE
G.f. = 1 - x^3 + x^4 - 2*x^5 + 2*x^6 - 3*x^7 + 4*x^8 - 5*x^9 + 6*x^10 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {x}, {x^2}, x^2, x^3], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=0, n, if( issquare( 3*k + 1, &m), (-1)^(m \ 3) * x^k ), A) / sum(k=0, (sqrtint(8*n + 1) - 1) \ 2, x^((k^2 + k) / 2), A), n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n+1) - 1, (-1)^k * x^(k^2 + 2*k) * prod(j=1, k, (1 - x^(2*j - 1)) / (1 - x^(2*j))^2, 1 + O(x^(n + 1 - k^2 - 2*k)))), n))};
CROSSREFS
Cf. A010054, A143064.
KEYWORD
sign
AUTHOR
Michael Somos, Jul 21 2008
STATUS
approved
A291316 Expansion of x/(1-x) + x^4*(1-x)/(1-x^3) + x^7*(1-x)*(1-x^3)/(1-x^5) + ... in powers of x. +10
0
1, 1, 1, 2, 0, 1, 3, -1, 1, 2, 0, 2, 1, 0, -1, 4, 2, -1, 2, -3, 4, 3, -1, 2, 0, 1, 1, 2, -2, 2, 5, 2, -3, 0, 1, -1, 6, 0, 4, -2, -1, 3, -1, 2, 0, 4, -2, 2, 4, -2, 1, 5, -2, -2, -2, 3, 6, 1, 3, -2, 4, -3, -1, -2, 3, 6, 2, 0, -4, 5, 1, 3, -1, 0, 0, 4, -1, -2, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Table of n, a(n) for n=1..79.
George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See equation (1.5)
FORMULA
a(n) = A008443(n) - A290735(n) = A290737(n) - A143064(n).
EXAMPLE
G.f. = x + x^2 + x^3 + 2*x^4 + x^6 + 3*x^7 - x^8 + x^9 + 2*x^10 + ...
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = sum(k=0, (n-1)\3, x^(3*k+1) * prod(i=1, k, 1 - x^(2*i-1), 1 + A) / (1 - x^(2*k+1)) ); polcoeff(A, n))};
CROSSREFS
Cf. A008443, A143064, A290735, A290737.
KEYWORD
sign
AUTHOR
Michael Somos, Aug 22 2017
STATUS
approved
A308745 Expansion of 1/(1 - x*(1 + x)/(1 - x^2*(1 + x^2)/(1 - x^3*(1 + x^3)/(1 - x^4*(1 + x^4)/(1 - ...))))), a continued fraction. +10
0
1, 1, 2, 4, 8, 17, 36, 76, 161, 342, 726, 1542, 3276, 6960, 14788, 31422, 66767, 141872, 301464, 640584, 1361188, 2892417, 6146164, 13060136, 27751818, 58970564, 125308114, 266270558, 565805452, 1202295228, 2554789536, 5428741218, 11535678790, 24512475453 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..33.
FORMULA
From Vaclav Kotesovec, Jun 25 2019: (Start)
a(n) ~ c * d^n, where
d = 2.124927028900893046638236231387101475346473032396641627320401...
c = 0.386397654364351443933577245182777062935616240164642598839093... (End)
From Peter Bala, Dec 18 2020
Conjectural g.f.: 1/(2 - (1 + x)/(1 - x^2/(2 - (1 + x^3)/(1 - x^4/(2 - (1 + x^5)/(1 - x^6/(2 - ... ))))))).
More generally it appears that 1/(1 - t*x*(1 + u*x)/(1 - t*x^2*(1 + u*x^2)/(1 - t*x^3*(1 + u*x^3)/(1 - t*x^4*(1 + u*x^4)/(1 - ... ))))) = 1/(1 + u - (u + t*x)/(1 - t*x^2/(1 + u - (u + t*x^3)/(1 - t*x^4/(1 + u - (u + t*x^5)/(1 - ... )))))). (End)
MATHEMATICA
nmax = 33; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^k (1 + x^k), 1, {k, 1, nmax}]), {x, 0, nmax}], x]
CROSSREFS
Cf. A005169, A053254, A092848, A143064.
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Jun 21 2019
STATUS
approved
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Last modified August 30 15:13 EDT 2024. Contains 375545 sequences. (Running on oeis4.)