A302450 Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(2*k^2-1)).

1, 1, 29, 182, 1084, 6593, 38878, 215937, 1169023, 6165895, 31737691, 159687840, 787536537, 3813036605, 18150405546, 85041775660, 392633910788, 1787993210106, 8037704764044, 35695268298904, 156708949403719, 680526030379206, 2924839092347883, 12447506657030287

Offset

0,3

Comments

Euler transform of A002593.

Links

Table of n, a(n) for n=0..23.

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

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A002593(k).

a(n) ~ exp(2^(5/3) * 3^(2/3) * Pi * n^(5/6) / (5 * 7^(1/6)) - Pi * sqrt(7*n) / 60 - 7^(7/6) * Pi * n^(1/6) / (1600 * 6^(2/3)) + Zeta(3) / (4*Pi^2) + 3*Zeta(5) / (2*Pi^4)) / (6^(2/3) * 7^(1/12) * n^(7/12)). - Vaclav Kotesovec, Apr 08 2018

Mathematica

nmax = 23; CoefficientList[Series[Product[1/(1 - x^k)^(k^2 (2 k^2 - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^3 (2 d^2 - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 23}]

Crossrefs

Cf. A002593, A023872, A287090, A302449.

Sequence in context: A373540 A140573 A165613 * A355623 A125339 A126497

Adjacent sequences: A302447 A302448 A302449 * A302451 A302452 A302453

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 08 2018

Status

approved