A192555 - OEIS

%I #14 Apr 19 2024 10:28:55

%S 1,0,2,18,302,7770,285182,14169498,916379102,74833699770,

%T 7532323742462,916288114073178,132533661862902302,

%U 22482642651307262970,4420834602574484743742,997471931914411955132058,255978001773528747607767902,74137405656663750753878861370

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

%C This sequence is the Akiyama-Tanigawa transform of the factorial numbers. - _Peter Luschny_, Apr 19 2024

%F a(n) = (-1)^n * Sum_{k=0..n} A163626(n, k)*k!. - _Philippe Deléham_, May 25 2015

%F a(n) ~ exp(-1/2) * n!^2. - _Vaclav Kotesovec_, Jul 05 2021

%p ATFactorial := proc(len)

%p local k, j, A, R, F; F := 1;

%p     for k from 0 to len do

%p         R[k] := F; F := F * (k + 1);

%p         for j from k by -1 to 1 do

%p             R[j - 1] := j * (R[j] - R[j-1])

%p         od;

%p         A[k] := R[0];

%p     od; convert(A, list) end:

%p ATFactorial(17);  # _Peter Luschny_, Apr 19 2024

%t Table[Sum[StirlingS2[n+1,k+1](-1)^(n-k)k!^2,{k,0,n}],{n,0,100}]

%o (Maxima) makelist(sum(stirling2(n+1,k+1)*(-1)^(n-k)*k!^2,k,0,n),n,0,24);

%Y Cf. A000142, A163626.

%K nonn

%O 0,3

%A _Emanuele Munarini_, Jul 04 2011