A295832 -id:A295832 - cp's OEIS frontend

A296045 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^n.

1, 1, 3, 13, 55, 231, 981, 4222, 18351, 80320, 353453, 1562364, 6932185, 30856541, 137725710, 616190583, 2762605791, 12408541299, 55825435656, 251523510045, 1134741006825, 5125453110196, 23175983361270, 104899547541255, 475228898015025, 2154737528486881, 9777332125043577

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A006950, A106337, A192540, A273225, A273226, A273228, A295832, A296043, A296044.

Table[SeriesCoefficient[Product[((1 + x^(2 k - 1))/(1 - x^(2 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^(4 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[(2 (-x)^(1/8)/EllipticTheta[2, 0, Sqrt[-x]])^n, {x, 0, n}], {n, 0, 26}]
Table[(-1)^n * 2^n * SeriesCoefficient[1/(QPochhammer[-1, x]*QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 07 2020 *)
(* Calculation of constants {d,c}: *) Chop[{1/r, 4/Sqrt[Pi*(77/2 - 4*s*(-r*s)^(7/8) * Derivative[0, 0, 2][EllipticTheta][2, 0, Sqrt[-r*s]])]} /. FindRoot[{s == (2*(-r*s)^(1/8))/EllipticTheta[2, 0, Sqrt[-r*s]], 7*I*r + 2*(-r*s)^(7/8)*Sqrt[r*s] * Derivative[0, 0, 1][EllipticTheta][2, 0, Sqrt[-r*s]] == 0}, {r, 1/5}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 - x^(4*k)))^n.

a(n) ~ c * d^n / sqrt(n), where d = 4.62579056836776492108784045382518984897... (see A192540) and c = 0.255113338880004277664416308115912337... - Vaclav Kotesovec, Dec 05 2017

A295831 Expansion of Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^k.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 19, 30, 47, 76, 118, 181, 277, 417, 624, 929, 1367, 2001, 2913, 4210, 6056, 8665, 12328, 17466, 24640, 34600, 48395, 67442, 93625, 129520, 178588, 245429, 336252, 459324, 625613, 849762, 1151150, 1555378, 2096332, 2818630, 3780903, 5060240, 6757633, 9005106, 11975265

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Cf. A001935, A001936, A001937, A026007, A035528, A093160, A113415, A156616, A284474, A292037, A295832.

nmax = 44; CoefficientList[Series[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 44; CoefficientList[Series[Exp[Sum[x^k (1 - (-1)^k x^k)/(k (1 - x^(2 k))^2), {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^k.
G.f.: exp(Sum_{k>=1} x^k*(1 - (-1)^k*x^k)/(k*(1 - x^(2*k))^2)).
a(n) ~ exp(3*(7*Zeta(3))^(1/3) * n^(2/3) / 4 + Pi^2 * n^(1/3) / (12 * (7*Zeta(3))^(1/3)) - Pi^4 / (3024*Zeta(3)) - 1/24) * A^(1/2) * (7*Zeta(3))^(11/72) / (2^(11/8) * sqrt(3*Pi) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017

A296046 Expansion of Product_{k>=1} ((1 - x^(2k))/(1 - x^(2k-1)))^k.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 3, 0, 3, 0, 6, 1, 3, 5, 0, 13, -3, 15, -3, 14, 6, 11, 16, -4, 38, -16, 51, -24, 65, -14, 46, 21, 10, 80, -49, 154, -102, 216, -136, 242, -119, 198, 1, 68, 189, -153, 486, -425, 775, -672, 1024, -779, 1035, -628, 782, -97, 96, 816, -930, 2069, -2203, 3428, -3413, 4546, -4130, 4958

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

sign

Cf. A008439, A008441, A008443, A010054, A295831, A295832, A296047.

nmax = 65; CoefficientList[Series[Product[((1 - x^(2 k))/(1 - x^(2 k - 1)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 - x^(2*k))/(1 - x^(2*k-1)))^k.

A296047 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 + x^(2k)))^k.

Original entry on oeis.org

1, 1, -1, 1, 1, 0, 0, -2, 5, 0, -2, 0, 3, 5, -11, 5, 6, 9, -17, -2, 23, -3, -11, -25, 62, -11, -27, -27, 76, 20, -104, 10, 77, 101, -243, 58, 118, 147, -353, -25, 378, 48, -372, -298, 892, -165, -444, -621, 1524, -128, -1055, -559, 1869, 575, -2682, 84, 2054, 1979, -5325, 844, 2947

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

sign

Cf. A029838, A029839, A029840, A029841, A029842, A029843, A029844, A029845, A295831, A295832, A296046.

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

G.f.: Product_{k>=1} ((1 + x^(2*k-1))/(1 + x^(2*k)))^k.

cp's OEIS Frontend

A296045 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^n.

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A295831 Expansion of Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^k.

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A296046 Expansion of Product_{k>=1} ((1 - x^(2k))/(1 - x^(2k-1)))^k.

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A296047 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 + x^(2k)))^k.

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cp's OEIS Frontend

A296045 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^n.

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A295831 Expansion of Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^k.

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A296046 Expansion of Product_{k>=1} ((1 - x^(2*k))/(1 - x^(2*k-1)))^k.

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A296047 Expansion of Product_{k>=1} ((1 + x^(2*k-1))/(1 + x^(2*k)))^k.

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A296045 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^n.

A295831 Expansion of Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^k.

A296046 Expansion of Product_{k>=1} ((1 - x^(2k))/(1 - x^(2k-1)))^k.

A296047 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 + x^(2k)))^k.