A349881 Expansion of Sum_{k>=0} x^k/(1 - k^4 * x).

1, 1, 2, 18, 339, 10915, 663140, 61264436, 8044351557, 1536980041573, 402558463751974, 137204787854813174, 60668198155262809815, 34351266752678243067591, 24185207999807747975188552, 20842786946335533698574605528

Offset: 0

Author

Seiichi Manyama, Dec 03 2021

Keywords

• nonn

Comments

In general, for t>=1 and s>=0, Sum_{k=0..n} k^(t*(n-k)+s) ~ sqrt(2*Pi) * ((n + s/t)/LambertW(exp(1)*(n + s/t)))^(1/2 + (t*n + s) * (1 - 1/LambertW(exp(1)*(n + s/t)))) / sqrt(t*(1 + LambertW(exp(1)*(n + s/t)))). - Vaclav Kotesovec, Dec 04 2021

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..195

Crossrefs

Cf. A026898, A234568, A349880.

Cf. A349858.

Programs

Mathematica

a[n_] := Sum[If[k == n - k == 0, 1, k^(4*(n - k))], {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Dec 04 2021 *)

PARI

a(n, s=0, t=4) = sum(k=0, n, k^(t*(n-k)+s));

PARI

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^4*x)))

Formula

a(n) = Sum_{k=0..n} k^(4*(n-k)).

a(n) ~ sqrt(Pi/2) * (n/LambertW(exp(1)*n))^(1/2 + 4*n - 4*n/LambertW(exp(1)*n)) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021