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A358687
a(n) = n! * Sum_{k=0..n} k^(3 * (n-k)) / (n-k)!.
3
1, 1, 4, 57, 1444, 61785, 4050126, 373648513, 47101090744, 7764843893265, 1630744323319450, 426925697290933401, 136591846585403311620, 52602030074554601172649, 24058544668572618782040022, 12916480280574798627072144465
OFFSET
0,3
LINKS
FORMULA
E.g.f.: Sum_{k>=0} x^k * exp(k^3 * x).
G.f.: Sum_{k>=0} k! * x^k / (1 - k^3 * x)^(k+1).
log(a(n)) ~ (6*n*(log(n) - 1) + 3*log(1 + LambertW(n^(2/3))) + 2*n*LambertW(n^(2/3)) * (7*log(n) - 6*log(1 + LambertW(n^(2/3))) + 3*LambertW(n^(2/3)))) / (6*(1 + LambertW(n^(2/3)))). - Vaclav Kotesovec, Nov 27 2022
MATHEMATICA
Join[{1}, Table[n! * Sum[k^(3*(n-k))/(n-k)!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 27 2022 *)
PROG
(PARI) a(n) = n!*sum(k=0, n, k^(3*(n-k))/(n-k)!);
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k*exp(x)^k^3)))
(PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*x^k/(1-k^3*x)^(k+1)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Nov 26 2022
STATUS
approved