A351137 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(3*n) * Stirling1(n,k).

1, 1, 129, 121172, 421875178, 3922823960054, 80130334773241142, 3156849112458066440568, 218554371053209725986724984, 24795129220015277612148345850896, 4365539219231132131300647267518575008, 1141930521329052244894253748456776246166288

Offset: 0

Author

Seiichi Manyama, Feb 02 2022

Keywords

• nonn

Comments

In general, for m >= 0, Sum_{k=0..n} (-1)^(n-k) * k! * k^(m*n) * Stirling1(n,k) ~ c * r^(m*n) * (1 + r*exp(m/r))^n * n^((m+1)*n + 1/2) / exp((m+1)*n), where r is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-m/r) and c is a constant (depending only on m). - Vaclav Kotesovec, Feb 04 2022

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..125

Crossrefs

Cf. A007840 (m=0), A320096 (m=1), A351136 (m=2).

Cf. A203798, A351134, A351138.

Programs

Mathematica

a[0] = 1; a[n_] := Sum[(-1)^(n - k) * k! * k^(3*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 12, 0] (* Amiram Eldar, Feb 02 2022 *)

PARI

a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^(3*n)*stirling(n, k, 1));

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k^3*x))^k)))

Formula

E.g.f.: Sum_{k>=0} (-log(1 - k^3*x))^k.

a(n) ~ c * r^(3*n) * (1 + r*exp(3/r))^n * n^(4*n + 1/2) / exp(4*n), where r = 0.97698437755148201976772582981871258235824532360125531194... is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-3/r) and c = 2.3655154360078103511101518906595610482889989819... - Vaclav Kotesovec, Feb 04 2022