A382737 a(n) = Sum_{k=0..n} k! * (k+1)! * Stirling2(n,k)^2.
1, 2, 14, 254, 8654, 467102, 36414734, 3862847774, 534433092494, 93409669590302, 20117959360842254, 5233190283794276894, 1617259866279958581134, 585633786711715561283102, 245587300036701328750786574, 118067003149791582488105955614, 64502003996859329263691323378574
Offset: 0
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 0..236
Programs
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Maple
f:= proc(n) local k; add(k!*(k+1)!*Stirling2(n,k)^2, k=0..n) end proc: map(f, [$0..40]);
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Mathematica
Table[Sum[k! * (k+1)! * StirlingS2[n,k]^2, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Apr 13 2025 *) -
PARI
a(n) = sum(k=0, n, k!*(k+1)!*stirling(n, k, 2)^2);
Formula
a(n) == 0 (mod 2) for n > 0.
a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x+y))^2.
a(n) ~ sqrt(Pi) * n^(2*n + 3/2) / (4 * sqrt(1 - log(2)) * exp(2*n) * log(2)^(2*n+2)). - Vaclav Kotesovec, Apr 13 2025