A302449 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(4*k^2-1)/3).

1, 1, 11, 46, 185, 700, 2676, 9646, 34166, 117500, 396506, 1310527, 4258313, 13607309, 42846151, 133039791, 407833188, 1235202869, 3699140386, 10960888382, 32154531807, 93437164720, 269087234273, 768340525743, 2176098269286, 6115444177489, 17058887661133

Offset: 0

Author

Ilya Gutkovskiy, Apr 08 2018

Keywords

• nonn

Comments

Euler transform of A000447.

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. A000335, A000447, A023871, A279215.

Programs

Mathematica

nmax = 26; CoefficientList[Series[Product[1/(1 - x^k)^(k (4 k^2 - 1)/3), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (4 d^2 - 1)/3, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 26}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A000447(k).

a(n) ~ exp(5 * Zeta(5)^(1/5) * n^(4/5)/2 - Zeta(3) * n^(2/5) / (12 * Zeta(5)^(2/5)) + 4*Zeta'(-3)/3 - 1/36 - Zeta(3)^2 / (720*Zeta(5))) * A^(1/3) * Zeta(5)^(83/900) / (2^(7/180) * sqrt(5*Pi) * n^(533/900)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2018