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A192555 a(n) = Sum_{k=0..n} Stirling2(n+1, k+1)*(-1)^(n-k)*k!^2.

%I A192555 #14 Apr 19 2024 10:28:55
%S A192555 1,0,2,18,302,7770,285182,14169498,916379102,74833699770,
%T A192555 7532323742462,916288114073178,132533661862902302,
%U A192555 22482642651307262970,4420834602574484743742,997471931914411955132058,255978001773528747607767902,74137405656663750753878861370
%N A192555 a(n) = Sum_{k=0..n} Stirling2(n+1, k+1)*(-1)^(n-k)*k!^2.
%C A192555 This sequence is the Akiyama-Tanigawa transform of the factorial numbers. - _Peter Luschny_, Apr 19 2024
%F A192555 a(n) = (-1)^n * Sum_{k=0..n} A163626(n, k)*k!. - _Philippe Deléham_, May 25 2015
%F A192555 a(n) ~ exp(-1/2) * n!^2. - _Vaclav Kotesovec_, Jul 05 2021
%p A192555 ATFactorial := proc(len)
%p A192555 local k, j, A, R, F; F := 1;
%p A192555     for k from 0 to len do
%p A192555         R[k] := F; F := F * (k + 1);
%p A192555         for j from k by -1 to 1 do
%p A192555             R[j - 1] := j * (R[j] - R[j-1])
%p A192555         od;
%p A192555         A[k] := R[0];
%p A192555     od; convert(A, list) end:
%p A192555 ATFactorial(17);  # _Peter Luschny_, Apr 19 2024
%t A192555 Table[Sum[StirlingS2[n+1,k+1](-1)^(n-k)k!^2,{k,0,n}],{n,0,100}]
%o A192555 (Maxima) makelist(sum(stirling2(n+1,k+1)*(-1)^(n-k)*k!^2,k,0,n),n,0,24);
%Y A192555 Cf. A000142, A163626.
%K A192555 nonn
%O A192555 0,3
%A A192555 _Emanuele Munarini_, Jul 04 2011