A295831 -id:A295831 - cp's OEIS frontend

A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^n.

1, 1, 5, 22, 101, 481, 2330, 11425, 56549, 281911, 1413465, 7120136, 36006362, 182681916, 929461993, 4740491107, 24229115109, 124069449335, 636376573943, 3268955179686, 16814509004601, 86593280920756, 446437797872016, 2303948443259841, 11900990745759578, 61526182236027756

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

nonn

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

Main diagonal of A296068.

Cf. A001935, A001936, A001937, A001939, A001940, A001941, A092877, A093160, A295831, A296043, A296045.

Table[SeriesCoefficient[Product[((1 + x^(2 k))/(1 - x^(2 k - 1)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[((1 - x^(4 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[(EllipticTheta[2, 0, x]/EllipticTheta[2, Pi/4, x^(1/2)]/(16 x)^(1/8))^n, {x, 0, n}], {n, 0, 25}]
(* Calculation of constant d: *) With[{k = 4}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

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

a(n) ~ c * d^n / sqrt(n), where d = 5.2749356339591798618290252741994029798069148326559... and c = 0.2726256757090475625917361066565981461455343437... - Vaclav Kotesovec, Dec 05 2017

A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^k.

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 12, 20, 33, 50, 74, 114, 175, 257, 375, 555, 814, 1171, 1677, 2406, 3435, 4855, 6825, 9591, 13428, 18667, 25851, 35745, 49250, 67544, 92340, 125966, 171345, 232257, 313945, 423470, 569778, 764465, 1023231, 1366827, 1821756, 2422394, 3214318, 4257088, 5627086, 7422941

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Cf. A000219, A006950, A113415, A156616, A224364, A263140, A273225, A273226, A274621, A295831.

nmax = 45; CoefficientList[Series[Product[((1 + x^(2 k - 1))/(1 - x^(2 k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[x^k ((-1)^(k + 1) + 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))/(1 - x^(2*k)))^k.
G.f.: exp(Sum_{k>=1} x^k*((-1)^(k+1) + 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) / (24 * (7*Zeta(3))^(1/3)) - Pi^4 / (12096 * Zeta(3)) + 1/12) * (7*Zeta(3))^(7/36) / (A * 2^(23/24) * sqrt(3*Pi) * n^(25/36)), 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

A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^n.

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A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^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

A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k))/(1 - x^(2*k-1)))^n.

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A295832 Expansion of Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^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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A296044 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k))/(1 - x^(2k-1)))^n.

A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^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.