OFFSET
0,3
COMMENTS
In general, for s>=1, Sum_{k=0..n} k^(n-k+s) ~ a(n) ~ sqrt(2*Pi) * ((n + s)/LambertW(exp(1)*(n + s)))^(1/2 + (n + s)*(1 - 1/LambertW(exp(1)*(n + s)))) / sqrt(1 + LambertW(exp(1)*(n + s))). - Vaclav Kotesovec, Dec 04 2021
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..500
FORMULA
a(n) = Sum_{k=0..n} k^(n-k+4).
a(n) ~ sqrt(2*Pi) * ((n + 4)/LambertW(exp(1)*(n + 4)))^(1/2 + (n + 4)*(1 - 1/LambertW(exp(1)*(n + 4)))) / sqrt(1 + LambertW(exp(1)*(n + 4))). - Vaclav Kotesovec, Dec 04 2021
MATHEMATICA
Table[Sum[k^(n - k + 4), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 04 2021 *)
PROG
(PARI) a(n, s=4, t=1) = sum(k=0, n, k^(t*(n-k)+s));
(PARI) my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^4*x^k/(1-k*x))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Dec 03 2021
STATUS
approved