A064898 Stirling transform of derangements numbers.

1, 0, 1, 5, 28, 199, 1721, 17394, 200803, 2607301, 37614922, 596933193, 10334308029, 193820343248, 3914731286181, 84716451763961, 1955520075368116, 47960724925499219, 1245468599978831333, 34139796082603477690, 985066290112167474255, 29844155285575945561913

Offset: 0

Author

Karol A. Penson, Oct 12 2001

Keywords

• nice
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..424
• H. Belbachir, Y. Djemmada and L. Németh, The deranged Bell numbers, arXiv:2102.00139 [math.GM], 2021.

Crossrefs

Cf. A000166.

Programs

Maple

g:= proc(n) option remember;
     `if`(n<2, 1-n, (n-1)*(g(n-1)+g(n-2)))
    end:
b:= proc(n, m) option remember;
     `if`(n=0, g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Feb 16 2023

Mathematica

A000166[n_] := Round[ n!/Exp[1] ]; A000166[0] = 1; A000166[1] = 0; a[n_] := Sum[ StirlingS2[n, k]*A000166[k], {k, 0, n}]; Table[ a[n], {n, 0, 18}] (* Jean-François Alcover, Dec 21 2011, after given formula *)

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k)*A000166(k).

E.g.f.: exp(-(exp(x)-1))/(2-exp(x)).

a(n) ~ n!/(2*exp(1)*log(2)^(n+1)). - Vaclav Kotesovec, Jun 29 2013