A248883 Expansion of Product_{k>=1} (1+x^k)^(k^4).

1, 1, 16, 97, 457, 2297, 11113, 52049, 235334, 1039886, 4497930, 19074006, 79418883, 325184763, 1311252535, 5212704708, 20449320159, 79231806015, 303428397505, 1149325838320, 4308477305997, 15993198330782, 58815616643170, 214383601754107, 774837953045873

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..3360 (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) ~ 31^(1/12) * exp(1/5 * (31/7)^(1/6) * 6^(2/3) * Pi * n^(5/6)) / (2^(7/6) * 3^(2/3) * 7^(1/12) * n^(7/12)).

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A284926(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(k*(1 - x^k)^5)). - Ilya Gutkovskiy, May 30 2018

Maple

b:= proc(n) option remember; add(
      (-1)^(n/d+1)*d^5, 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^4), {k, 1, nmax}], {x, 0, nmax}], x]

Prog

(PARI) x = 'x + O('x^50); Vec(prod(k=1, 50, (1 + x^k)^(k^4))) \\ Indranil Ghosh, Apr 06 2017
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^4: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

Crossrefs

Cf. A026007, A027998, A248882, A248884.

Column k=4 of A284992.

Sequence in context: A041488 A277225 A265841 * A223902 A264580 A122102

Adjacent sequences: A248880 A248881 A248882 * A248884 A248885 A248886

Keyword

nonn

Author

Vaclav Kotesovec, Mar 05 2015

Status

approved