OFFSET
0,3
COMMENTS
For m>0, Sum_{k=0..n} binomial(n,k)^m * k^n ~ c(m) * d^n * n! / n^(m/2), where d = (1 + m*LambertW(exp(-1/m)/m))^(m-1) / (m^m * LambertW(exp(-1/m)/m)^m) and c(m) is a constant independent of n.
FORMULA
a(n) ~ c * d^n * n! / n^(3/2), where d = (1 + 3*LambertW(exp(-1/3)/3))^2 / (27 * LambertW(exp(-1/3)/3)^3) = 12.3645613141726293982008517178673172577947617775... and c = 0.143687082995832067469009730530027989920523409582173778129054767279...
MATHEMATICA
Join[{1}, Table[Sum[k^n * Binomial[n, k]^3, {k, 0, n}], {n, 1, 20}]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Feb 20 2021
STATUS
approved