A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).

1, 1, 8, 35, 119, 433, 1476, 4962, 16128, 51367, 160105, 490219, 1476420, 4378430, 12805008, 36962779, 105417214, 297265597, 829429279, 2291305897, 6270497702, 17008094490, 45744921052, 122052000601, 323166712109, 849453194355, 2217289285055, 5749149331789

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..5510 (terms 0..1000 from Vaclav Kotesovec)

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 22.

Formula

a(n) ~ Zeta(5)^(1/10) * 3^(1/5) * exp(2^(-11/5) * 3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) * n^(4/5)) / (2^(71/120) * 5^(2/5)* sqrt(Pi) * n^(3/5)), where Zeta(5) = A013663.

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A284900(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 4*x^k + x^(2*k))/(k*(1 - x^k)^4)). - Ilya Gutkovskiy, May 30 2018

Euler transform of A309335. - Georg Fischer, Nov 10 2020

Maple

b:= proc(n) option remember; add(
      (-1)^(n/d+1)*d^4, d=numtheory[divisors](n))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 16 2017

Mathematica

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k^3), {k, 1, nmax}], {x, 0, nmax}], x]

Prog

(PARI) x = 'x + O('x^50); Vec(prod(k=1, 50, (1 + x^k)^(k^3))) \\ Indranil Ghosh, Apr 06 2017
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^3: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

Crossrefs

Cf. A026007, A027998, A248883, A248884, A309335.

Column k=3 of A284992.

Sequence in context: A036598 A229403 A059824 * A292479 A301881 A094616

Adjacent sequences: A248879 A248880 A248881 * A248883 A248884 A248885

Keyword

nonn

Author

Vaclav Kotesovec, Mar 05 2015

Status

approved