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%I #9 Jul 05 2021 07:13:17
%S 1,1,3,25,387,9481,336723,16340185,1038177507,83616187561,
%T 8323660051443,1003415542660345,144043181112445827,
%U 24279259683302736841,4747993384270354742163,1066206704980940216628505,272480888391150986151565347
%N a(n) = sum(stirling2(n,k)*(-1)^(n-k)*k!^2,k=0..n).
%F O.g.f.: Sum_{n>=0} n!^2 * x^n / Product_{k=0..n} (1 + k*x). [From Paul D. Hanna, Jul 20 2011]
%F a(n) ~ exp(-1/2) * n!^2. - _Vaclav Kotesovec_, Jul 05 2021
%t Table[Sum[StirlingS2[n,k](-1)^(n-k)k!^2,{k,0,n}],{n,0,100}]
%o (Maxima) makelist(sum(stirling2(n,k)*(-1)^(n-k)*k!^2,k,0,n),n,0,24);
%o (PARI) {a(n)=polcoeff(sum(m=0, n, m!^2*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */
%K nonn
%O 0,3
%A _Emanuele Munarini_, Jul 04 2011
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