A242280 a(n) = Sum_{k=0..n} (k! * Stirling2(n,k))^3.

1, 1, 9, 433, 63225, 18954001, 10159366329, 8924902306993, 11969476975085625, 23232038620328946001, 62655369716047066046649, 227268291642918880258797553, 1079475019974966974009683584825, 6565863403062578428919598754170001

Offset: 0

Author

Vaclav Kotesovec, May 10 2014

Keywords

• nonn
• easy

Comments

Generally, for p>=1 is Sum_{k=0..n} (k!*StirlingS2(n,k))^p asymptotic to n^(p*n+1/2) * sqrt(Pi/(2*p*(1-log(2))^(p-1))) / (exp(p*n) * log(2)^(p*n+1)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..169

Crossrefs

Cf. A000670, A048144.

Programs

Mathematica

Table[Sum[(k!)^3 * StirlingS2[n,k]^3,{k,0,n}],{n,0,20}]

PARI

a(n) = sum(k=0, n, (k!*stirling(n, k, 2))^3); \\ Seiichi Manyama, Apr 06 2025

Formula

a(n) ~ sqrt(Pi/6) * n^(3*n+1/2) / ((1-log(2)) * exp(3*n) * (log(2))^(3*n+1)).

a(n) = (n!)^3 * [(x*y*z)^n] 1 / (1 - (exp(x) - 1) * (exp(y) - 1) * (exp(z) - 1)). - Seiichi Manyama, Apr 06 2025