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

1, 0, 2, 18, 302, 7770, 285182, 14169498, 916379102, 74833699770, 7532323742462, 916288114073178, 132533661862902302, 22482642651307262970, 4420834602574484743742, 997471931914411955132058, 255978001773528747607767902, 74137405656663750753878861370

Offset: 0

Author

Emanuele Munarini, Jul 04 2011

Keywords

• nonn

Comments

This sequence is the Akiyama-Tanigawa transform of the factorial numbers. - Peter Luschny, Apr 19 2024

Crossrefs

Cf. A000142, A163626.

Programs

Maple

ATFactorial := proc(len)
local k, j, A, R, F; F := 1;
    for k from 0 to len do
        R[k] := F; F := F * (k + 1);
        for j from k by -1 to 1 do
            R[j - 1] := j * (R[j] - R[j-1])
        od;
        A[k] := R[0];
    od; convert(A, list) end:
ATFactorial(17);  # Peter Luschny, Apr 19 2024

Mathematica

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

Maxima

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

Formula

a(n) = (-1)^n * Sum_{k=0..n} A163626(n, k)*k!. - Philippe Deléham, May 25 2015

a(n) ~ exp(-1/2) * n!^2. - Vaclav Kotesovec, Jul 05 2021