A302447 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)^2/2).

1, 2, 12, 46, 175, 610, 2107, 6918, 22256, 69498, 212649, 636910, 1874470, 5423332, 15457223, 43433088, 120467606, 330077358, 894193347, 2396636236, 6359325300, 16714566278, 43539016461, 112449776138, 288083439729, 732356943548, 1848098069644, 4630892393996

Offset

0,2

Comments

Euler transform of A006002.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..6000

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)^A006002(k).

a(n) ~ exp(5 * (3*Zeta(5))^(1/5) * n^(4/5) / 2^(8/5) + Pi^4 * n^(3/5) / (90 * 2^(1/5) * (3*Zeta(5))^(3/5)) + (Zeta(3) / 2^(9/5) - Pi^8 / (27000 * 2^(4/5) * Zeta(5))) * n^(2/5) / (3*Zeta(5))^(2/5) + (Pi^8 / (12150000 * Zeta(5)) - Zeta(3) / 900) * Pi^4 * n^(1/5) / (2^(2/5) * 3^(1/5) * Zeta(5)^(6/5)) + 1/24 - Zeta(3) / (4*Pi^2) - Pi^16 / (5248800000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (324000 * Zeta(5)^2) - Zeta(3)^2 / (120 * Zeta(5)) + Zeta'(-3)/2) * (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 = 27; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1)^2/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1)^2/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]

Crossrefs

Cf. A000294, A006002, A294667.

Sequence in context: A006742 A003993 A129018 * A319763 A320684 A069946

Adjacent sequences: A302444 A302445 A302446 * A302448 A302449 A302450

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 08 2018

Status

approved