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