A295828 Expansion of Product_{k>=1} 1/(1 - x^k)^(2*k*(2*k-1)).

1, 2, 15, 58, 235, 862, 3122, 10664, 35639, 115164, 363806, 1122050, 3393316, 10068006, 29374056, 84347944, 238713339, 666419456, 1836986443, 5003473866, 13476019215, 35912177618, 94746481999, 247597696802, 641205816641, 1646268490598, 4192059724668, 10590937903412, 26556243826240

Offset: 0

Author

Ilya Gutkovskiy, Nov 28 2017

Keywords

• nonn

Comments

Euler transform of A002939.

Links

• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A002939, A161870, A253289, A258347, A258348, A278767.

Programs

Mathematica

nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^(2 k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[2 d^2 (2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A002939(k).

a(n) ~ exp(2^(5/2) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - Zeta(3) * sqrt(15*n) / Pi^2 - 15^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(3/2) * Pi^5) - Zeta(3) / Pi^2 - 75*Zeta(3)^3 / (2*Pi^8) - 1/6) * A^2 / (2^(4/3) * 15^(1/12) * Pi^(1/6) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017