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A349881
Expansion of Sum_{k>=0} x^k/(1 - k^4 * x).
4
1, 1, 2, 18, 339, 10915, 663140, 61264436, 8044351557, 1536980041573, 402558463751974, 137204787854813174, 60668198155262809815, 34351266752678243067591, 24185207999807747975188552, 20842786946335533698574605528
OFFSET
0,3
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
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
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 *)
PROG
(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)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Dec 03 2021
STATUS
approved