A273226 -id:A273226 - 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

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

A273228 G.f. is the fourth power of the g.f. of A006950.

Original entry on oeis.org

1, 4, 10, 24, 55, 116, 230, 440, 819, 1480, 2602, 4480, 7580, 12604, 20620, 33272, 53029, 83520, 130088, 200600, 306488, 464168, 697150, 1039032, 1537435, 2259300, 3298428, 4785880, 6903657, 9903040, 14129846, 20058488, 28336790, 39845456, 55778050, 77747328, 107924347, 149221160

Offset: 0

M.S. Mahadeva Naika, May 18 2016

nonn

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), 273--284.
M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
L. Wang, Arithmetic properties of partition triples with odd parts distinct, Int. J. Number Theory, 11 (2015), 1791--1805.
L. Wang, Arithmetic properties of partition quadruples with odd parts distinct, Bull. Aust. Math. Soc., doi:10.1017/S0004972715000647.
L. Wang, New congruences for partitions where the odd parts are distinct, J. Integer Seq. (2015), article 15.4.2.

Cf. A006960, A273226.

Digits:=200:with(PolynomialTools): with(qseries): with(ListTools):
GenFun:=series(etaq(q,2,1000)^4/etaq(q,1,1000)^4/etaq(q,4,1000)^4,q,50):
CoefficientList(sort(convert(GenFun,polynom),q,ascending),q);

nmax = 30; CoefficientList[Series[Product[(1 + x^k)^4 / (1 - x^(4*k))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 25 2017 *)
CoefficientList[Series[1/(QPochhammer[q, -q]*QPochhammer[q^2, q^2])^4, {q, 0, 50}], q] (* G. C. Greubel, Apr 17 2018 *)

G.f.: Product_{k>=1} (1 + x^k)^4 / (1 - x^(4*k))^4, corrected by Vaclav Kotesovec, Mar 25 2017
Expansion of 1 / psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.
a(n) ~ exp(sqrt(2*n)*Pi) / (2^(9/4)*n^(7/4)). - Vaclav Kotesovec, Mar 25 2017

Edited by N. J. A. Sloane, May 26 2016

cp's OEIS Frontend

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

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

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A273228 G.f. is the fourth power of the g.f. of A006950.

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

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A273228 G.f. is the fourth power of the g.f. of A006950.

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Author

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Programs

Maple

Mathematica

Formula

Extensions

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

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