A302448 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k^2+1)/2).

1, 1, 6, 21, 70, 210, 646, 1881, 5446, 15295, 42355, 115036, 308312, 814023, 2123431, 5471967, 13949888, 35194914, 87952796, 217803302, 534794576, 1302545064, 3148316746, 7554386885, 18001627175, 42613759083, 100240372671, 234371794954, 544812235887

Offset

0,3

Comments

Euler transform of A006003.

Links

Table of n, a(n) for n=0..28.

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)^A006003(k).

a(n) ~ exp(5 * (3*Zeta(5))^(1/5) * n^(4/5) / 2^(8/5) + Zeta(3) * n^(2/5) / (2^(9/5) * (3*Zeta(5))^(2/5)) + Zeta'(-3)/2 + 1/24 - Zeta(3)^2 / (120 * Zeta(5))) * (3*Zeta(5))^(43/400) / (2^(57/200) * sqrt(5*A*Pi) * n^(243/400)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2018

Mathematica

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

Crossrefs

Cf. A000294, A006003, A295179, A302447.

Sequence in context: A318943 A200761 A169687 * A101904 A022814 A000390

Adjacent sequences: A302445 A302446 A302447 * A302449 A302450 A302451

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 08 2018

Status

approved