A382853 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * (k! * Stirling1(n,k))^2.
1, 1, 14, 588, 51064, 7542780, 1688795184, 532244030976, 224335607135616, 121793234373123840, 82750681453274478720, 68773648886955417943296, 68628724852793337500166144, 80970628401965472953705395200, 111490683570184861858636405923840, 177177650274516448010905794637332480
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[Binomial[n+k-1,k] * k!^2 * StirlingS1[n,k]^2, {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Apr 07 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(n+k-1, k)*(k!*stirling(n, k, 1))^2);
Formula
a(n) == 0 (mod n) for n > 0.
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1-x) * log(1-y))^n.
a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1+x) * log(1+y))^n.
a(n) ~ c * (r*(1+r) + sqrt(r*(1+r)))^(2*n) * n^(2*n) / (exp(2*n) * r^n), where r = 0.71197519729041875298209529969157574831688314013967... is the root of the equation (1+r)*(r + LambertW(-1, -r*exp(-r)))^2 = r and c = 0.61294561390083215776201123658816241786650851195222... - Vaclav Kotesovec, Apr 07 2025