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
Keywords
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
Programs
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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}]
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
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