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

0,4

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.

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

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 *)

Crossrefs

Cf. A000166.

Sequence in context: A003467 A240770 A318364 * A368792 A216586 A151500

Adjacent sequences: A064895 A064896 A064897 * A064899 A064900 A064901

Keyword

nice,nonn

Author

Karol A. Penson, Oct 12 2001

Status

approved