A007096 -id:A007096 - cp's OEIS frontend

A261648 Expansion of Product_{k>=0} ((1+x^(2k+1))/(1-x^(2k+1)))^5.

1, 10, 50, 180, 550, 1512, 3820, 9040, 20310, 43670, 90472, 181540, 354180, 674040, 1254640, 2289104, 4101430, 7228020, 12546030, 21473940, 36281656, 60565920, 99974140, 163297520, 264110180, 423211938, 672244600, 1059013320, 1655274320, 2568068120

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

In general, if j > 0 and g.f. = Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^j, then a(n) ~ exp(Pi*sqrt(j*n/2)) * j^(1/4) / (2^(j/2 + 7/4) * n^(3/4)).

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 11.

Cf. A080054 (j=1), A007096 (j=2), A261647 (j=3), A014969 (j=4), A014970 (j=6), A014972 (j=8), A103261 (j=10).

nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^5,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(5*n/2)) * 5^(1/4) / (16 * 2^(1/4) * n^(3/4)).

A320967 Expansion of Product_{k>0} theta_3(q^k)/theta_4(q^k), where theta_3() and theta_4() are the Jacobi theta functions.

Original entry on oeis.org

1, 4, 12, 36, 92, 220, 508, 1108, 2332, 4776, 9492, 18420, 35036, 65324, 119708, 216044, 384204, 674236, 1168968, 2003460, 3397300, 5704148, 9487740, 15642676, 25577900, 41495032, 66817812, 106837112, 169677372, 267755836, 419948980, 654799316, 1015276412, 1565765892

Offset: 0

Seiichi Manyama, Oct 25 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Self-convolution of A320968.

Cf. A000122, A002448, A007096, A301554, A320067, A320970.

With[{nmax=50}, CoefficientList[Series[Product[EllipticTheta[3, 0, q^k]/EllipticTheta[4, 0, q^k], {k, 1, nmax+2}], {q, 0, nmax}], q]] (* G. C. Greubel, Oct 29 2018 *)

m=50; q='q+O('q^m); Vec(prod(k=1,m+2, eta(q^(2*k))^6/(eta(q^k)^4* eta(q^(4*k))^2) )) \\ G. C. Greubel, Oct 29 2018

Expansion of Product_{k>0} eta(q^(2*k))^6 / (eta(q^k)^4*eta(q^(4*k))^2).

A261649 Expansion of Product_{k>=0} ((1+x^(3k+1))/(1-x^(3k+1)))^2.

Original entry on oeis.org

1, 4, 8, 12, 20, 36, 56, 80, 120, 180, 252, 348, 492, 680, 912, 1228, 1652, 2180, 2856, 3744, 4860, 6256, 8044, 10284, 13048, 16520, 20848, 26140, 32672, 40756, 50596, 62576, 77256, 95060, 116540, 142592, 174036, 211736, 257056, 311448, 376332, 453764, 546160

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

Cf. A015128, A156616, A080054, A007096, A261610, A261651.

nmax=60; CoefficientList[Series[Product[((1+x^(3*k+1))/(1-x^(3*k+1)))^2,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(2*n/3)) * Gamma(1/3)^2 / (2^(7/4) * 3^(5/12) * Pi^(4/3) * n^(7/12)).

A261651 Expansion of Product_{k>=0} ((1+x^(3k+1))/(1-x^(3k+1)))^3.

Original entry on oeis.org

1, 6, 18, 38, 72, 138, 254, 432, 708, 1154, 1836, 2826, 4288, 6456, 9552, 13902, 20070, 28722, 40614, 56916, 79242, 109448, 149988, 204318, 276672, 372288, 498264, 663602, 879252, 1159470, 1522564, 1990788, 2592162, 3362638, 4346244, 5597100, 7183792, 9191004

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

Cf. A015128, A156616, A080054, A007096, A261610, A261649.

nmax=60; CoefficientList[Series[Product[((1+x^(3*k+1))/(1-x^(3*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(n)) * Gamma(1/3)^3 / (4 * Pi^2 * sqrt(3*n)).

A261652 Expansion of Product_{k>=0} ((1+x^(4k+1))/(1-x^(4k+1)))^3.

Original entry on oeis.org

1, 6, 18, 38, 66, 108, 182, 306, 486, 728, 1068, 1578, 2318, 3312, 4614, 6388, 8862, 12192, 16488, 22038, 29400, 39156, 51702, 67554, 87810, 113982, 147384, 189200, 241446, 307356, 390408, 493662, 621006, 778712, 974628, 1216284, 1511756, 1872840, 2315538

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} ((1 + x^(a*k+b))/(1 - x^(a*k+b)))^j, then a(n) ~ Gamma(b/a)^j * 2^(j/2 - 3/2 - 2*b*j/a) * a^(-j/4 - 1/4 + b*j/(2*a)) * exp(Pi*sqrt(j*n/a)) * j^(1/4 - j/4 + b*j/(2*a)) * Pi^(b*j/a - j) * n^(j/4 - 3/4 - b*j/(2*a)).

Cf. A015128 (a=1, b=1, j=1), A156616.
Cf. A080054 (a=2, b=1, j=1), A007096 (a=2, b=1, j=2), A261647 (a=2, b=1, j=3), A014969 (a=2, b=1, j=4), A261648 (a=2, b=1, j=5), A014970 (a=2, b=1, j=6), A014972 (a=2, b=1, j=8), A103261 (a=2, b=1, j=10).
Cf. A261610 (a=3, b=1, j=1), A261649 (a=3, b=1, j=2), A261651 (a=3, b=1, j=3).
Cf. A261611 (a=4, b=1, j=1), A261650 (a=4, b=1, j=2), A261652 (a=4, b=1, j=3).

nmax=60; CoefficientList[Series[Product[((1+x^(4*k+1))/(1-x^(4*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(3*n)/2) * 2^(1/4) * Gamma(1/4)^3 / (8 * 3^(1/8) * Pi^(9/4) * n^(3/8)).

A261650 Expansion of Product_{k>=0} ((1+x^(4k+1))/(1-x^(4k+1)))^2.

Original entry on oeis.org

1, 4, 8, 12, 16, 24, 40, 60, 80, 104, 144, 204, 272, 344, 440, 584, 768, 968, 1200, 1516, 1936, 2424, 2968, 3644, 4528, 5596, 6800, 8216, 10000, 12184, 14688, 17564, 21056, 25320, 30272, 35912, 42576, 50616, 60024, 70728, 83136, 97896, 115200, 134924, 157504

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

Cf. A015128, A156616, A080054, A007096, A261611, A261652.

nmax=60; CoefficientList[Series[Product[((1+x^(4*k+1))/(1-x^(4*k+1)))^2,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(n/2)) * 2^(3/2) * Gamma(1/4)^2 / (16 * Pi^(3/2) * sqrt(n)).

A289522 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=0} ((1 + x^(2j+1))/(1 - x^(2j+1)))^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 2, 0, 1, 6, 8, 4, 0, 1, 8, 18, 16, 6, 0, 1, 10, 32, 44, 32, 8, 0, 1, 12, 50, 96, 102, 56, 12, 0, 1, 14, 72, 180, 256, 216, 96, 16, 0, 1, 16, 98, 304, 550, 624, 428, 160, 22, 0, 1, 18, 128, 476, 1056, 1512, 1408, 816, 256, 30, 0, 1, 20, 162, 704, 1862, 3240, 3820, 3008, 1494, 404, 40, 0

Offset: 0

Ilya Gutkovskiy, Jul 07 2017

			Square array begins:
1,  1,   1,    1,    1,     1,  ...
0,  2,   4,    6,    8,    10,  ...
0,  2,   8,   18,   32,    50,  ...
0,  4,  16,   44,   96,   180,  ...
0,  6,  32,  102,  256,   550,  ...
0,  8,  56,  216,  624,  1512,  ...

Columns k=0-6 give: A000007, A080054, A007096, A261647, A014969, A261648, A014970.
Rows n=0-3 give: A000012, A005843, A001105, A217873.
Main diagonal gives A291697.

Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i + 1))/(1 - x^(2 i + 1)))^k, {i, 0, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=0} ((1 + x^(2*j+1))/(1 - x^(2*j+1)))^k.
G.f. of column 2k: (theta_3(x)/theta_4(x))^k, where theta_() is the Jacobi theta function.
For asymptotics of column k see comment from Vaclav Kotesovec in A261648.

A308288 Expansion of Product_{i>=1, j>=1} theta_3(x^(ij))/theta_4(x^(ij)), where theta_() is the Jacobi theta function.

Original entry on oeis.org

1, 4, 16, 56, 172, 496, 1360, 3528, 8824, 21344, 50048, 114360, 255336, 557888, 1195952, 2519264, 5221076, 10660512, 21467904, 42674520, 83812560, 162753584, 312689168, 594740456, 1120498048, 2092059800, 3872731232, 7110830376, 12955269304, 23428775520

Offset: 0

Ilya Gutkovskiy, May 18 2019

nonn

Convolution of the sequences A305050 and A308286.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A000005, A000122, A002448, A007096, A007425, A301554, A305050, A308286, A320967, A320970.

nmax = 29; CoefficientList[Series[Product[Product[EllipticTheta[3, 0, x^(i j)]/EllipticTheta[4, 0, x^(i j)], {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[Product[(EllipticTheta[3, 0, x^k]/EllipticTheta[4, 0, x^k])^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (theta_3(x^k)/theta_4(x^k))^tau(k), where tau = number of divisors (A000005).
G.f.: Product_{i>=1, j>=1} (Sum_{k=-oo..+oo} x^(i*j*k^2))/(Sum_{k=-oo..+oo} (-1)^k*x^(i*j*k^2)).
G.f.: Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^4/(1 + x^(2*i*j*k))^2.
G.f.: Product_{k>=1} (1 + x^k)^(4*tau_3(k))/(1 + x^(2*k))^(2*tau_3(k)), where tau_3 = A007425.

A294592 a(n) = [x^n] (theta_3(x)/theta_4(x))^n, where theta_() is the Jacobi theta function.

Original entry on oeis.org

1, 4, 32, 304, 3072, 32024, 340352, 3666016, 39878656, 437091892, 4819567552, 53401892240, 594093969408, 6631726263608, 74242911364864, 833237193123104, 9371924860764160, 105614054423502408, 1192210691317862048, 13478559927485340144, 152589996020498655232, 1729590806617202662528

Offset: 0

Ilya Gutkovskiy, Nov 03 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..500
Eric Weisstein's MathWorld, Jacobi Theta Functions
Wikipedia, Theta function

Cf. A007096, A014969, A014970, A014972, A066535, A080054, A103261, A270919, A289522, A291697.

S:= series((JacobiTheta3(0,x)/JacobiTheta4(0,x))^n,x,51):
seq(coeff(S,x,n),n=0..50); # Robert Israel, Nov 03 2017

Table[SeriesCoefficient[(EllipticTheta[3, 0, x]/EllipticTheta[4, 0, x])^n, {x, 0, n}], {n, 0, 21}]
Table[SeriesCoefficient[Product[((1 + x^(2 k + 1))/(1 - x^(2 k + 1)))^(2 n), {k, 0, n}], {x, 0, n}], {n, 0, 21}]
Table[SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^(2 n), {x, 0, n}], {n, 0, 21}]
(* Calculation of constant d: *) 1/r /. FindRoot[{s == EllipticTheta[3, 0, r*s]/EllipticTheta[4, 0, r*s], EllipticTheta[4, 0, r*s] + r*s*Derivative[0, 0, 1][EllipticTheta][4, 0, r*s] == r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s]}, {r, 1/10}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^(2*n).
From Vaclav Kotesovec, Nov 05 2017: (Start)
a(n) ~ c * d^n / sqrt(n), where
d = 11.61255065799699699891360038489317237925475956178123836149123386457... and
c = 0.34456510029264878768512693687607064416428117641473856418257649837... (End)

A326827 Expansion of 1 / (chi(-x)^10 * chi(-x^2)^4) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 10, 59, 270, 1045, 3582, 11194, 32488, 88716, 230150, 571363, 1365148, 3153522, 7069242, 15425719, 32849906, 68421073, 139645914, 279740407, 550790788, 1067244261, 2037348726, 3835457084, 7126887974, 13081454919, 23735283778, 42598577587, 75668099822

Offset: 0

Michael Somos, Oct 20 2019

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 10*x + 59*x^2 + 270*x^3 + 1045*x^4 + 3582*x^5 + 11194*x^6 + ...
G.f. = q^3 + 10*q^7 + 59*q^11 + 270*q^15 + 1045*q^19 + 3582*q^23 + 11194*q^27 + ...

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A007096, A093160, A123655, A189925, A212318, A215348, A215349, A232358, A232772.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 QPochhammer[ x^4]^2 / (QPochhammer[ x]^5))^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ x^(-3/4) (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0, x]^2 / 4)^2, {x, 0, n}];
nmax = 20; CoefficientList[Series[Product[(1 + x^k)^10/(1 - x^(4*k - 2))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 31 2019 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 * eta(x^4 + A)^2 / eta(x + A)^5)^2, n))};

Expansion of q^(-3/4) * (eta(q^2)^3 * eta(q^4)^2 / eta(q)^5)^2 in powers of q.
Euler transform of period 4 sequence [10, 4, 10, 0, ...].
G.f.: Product_{n>=0} (1 - x^(2*n + 1))^-10 * (1 - x^(4*n + 2))^-4.
A093160(2*n + 1) = A123655(4*n + 3) = 4*a(n).
A232772(2*n + 1) = A215348(4*n + 3) = A215349(4*n + 3) = 8*a(n).
A007096(4*n + 3) = A212318(4*n + 3) = 16*a(n). A189925(4*n + 3) = A232358(4*n + 3) = -16*a(n).
a(n) ~ exp(2*Pi*sqrt(n)) / (256*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019

cp's OEIS Frontend

A261648 Expansion of Product_{k>=0} ((1+x^(2*k+1))/(1-x^(2*k+1)))^5.

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A320967 Expansion of Product_{k>0} theta_3(q^k)/theta_4(q^k), where theta_3() and theta_4() are the Jacobi theta functions.

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A261649 Expansion of Product_{k>=0} ((1+x^(3*k+1))/(1-x^(3*k+1)))^2.

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A261651 Expansion of Product_{k>=0} ((1+x^(3*k+1))/(1-x^(3*k+1)))^3.

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A261652 Expansion of Product_{k>=0} ((1+x^(4*k+1))/(1-x^(4*k+1)))^3.

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A261650 Expansion of Product_{k>=0} ((1+x^(4*k+1))/(1-x^(4*k+1)))^2.

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A289522 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=0} ((1 + x^(2*j+1))/(1 - x^(2*j+1)))^k.

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A308288 Expansion of Product_{i>=1, j>=1} theta_3(x^(i*j))/theta_4(x^(i*j)), where theta_() is the Jacobi theta function.

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A294592 a(n) = [x^n] (theta_3(x)/theta_4(x))^n, where theta_() is the Jacobi theta function.

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A261648 Expansion of Product_{k>=0} ((1+x^(2k+1))/(1-x^(2k+1)))^5.

A261649 Expansion of Product_{k>=0} ((1+x^(3k+1))/(1-x^(3k+1)))^2.

A261651 Expansion of Product_{k>=0} ((1+x^(3k+1))/(1-x^(3k+1)))^3.

A261652 Expansion of Product_{k>=0} ((1+x^(4k+1))/(1-x^(4k+1)))^3.

A261650 Expansion of Product_{k>=0} ((1+x^(4k+1))/(1-x^(4k+1)))^2.

A289522 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=0} ((1 + x^(2j+1))/(1 - x^(2j+1)))^k.

A308288 Expansion of Product_{i>=1, j>=1} theta_3(x^(ij))/theta_4(x^(ij)), where theta_() is the Jacobi theta function.