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A064898 Stirling transform of derangements numbers.

%I A064898 #28 Feb 16 2023 15:14:17
%S A064898 1,0,1,5,28,199,1721,17394,200803,2607301,37614922,596933193,
%T A064898 10334308029,193820343248,3914731286181,84716451763961,
%U A064898 1955520075368116,47960724925499219,1245468599978831333,34139796082603477690,985066290112167474255,29844155285575945561913
%N A064898 Stirling transform of derangements numbers.
%H A064898 Alois P. Heinz, <a href="/A064898/b064898.txt">Table of n, a(n) for n = 0..424</a>
%H A064898 H. Belbachir, Y. Djemmada and L. Németh, <a href="https://arxiv.org/abs/2102.00139">The deranged Bell numbers</a>, arXiv:2102.00139 [math.GM], 2021.
%F A064898 a(n) = Sum_{k=0..n} Stirling2(n,k)*A000166(k).
%F A064898 E.g.f.: exp(-(exp(x)-1))/(2-exp(x)).
%F A064898 a(n) ~ n!/(2*exp(1)*log(2)^(n+1)). - _Vaclav Kotesovec_, Jun 29 2013
%p A064898 g:= proc(n) option remember;
%p A064898      `if`(n<2, 1-n, (n-1)*(g(n-1)+g(n-2)))
%p A064898     end:
%p A064898 b:= proc(n, m) option remember;
%p A064898      `if`(n=0, g(m), m*b(n-1, m)+b(n-1, m+1))
%p A064898     end:
%p A064898 a:= n-> b(n, 0):
%p A064898 seq(a(n), n=0..27);  # _Alois P. Heinz_, Feb 16 2023
%t A064898 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 *)
%Y A064898 Cf. A000166.
%K A064898 nice,nonn
%O A064898 0,4
%A A064898 _Karol A. Penson_, Oct 12 2001