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A171367 Antidiagonal sums of triangle of Stirling numbers of 2nd kind A048993.

%I A171367 #40 Jan 24 2025 17:11:42
%S A171367 1,0,1,1,2,4,9,22,58,164,495,1587,5379,19195,71872,281571,1151338,
%T A171367 4902687,21696505,99598840,473466698,2327173489,11810472444,
%U A171367 61808852380,333170844940,1847741027555,10532499571707,61649191750137,370208647200165,2278936037262610,14369780182166215
%N A171367 Antidiagonal sums of triangle of Stirling numbers of 2nd kind A048993.
%H A171367 Seiichi Manyama, <a href="/A171367/b171367.txt">Table of n, a(n) for n = 0..665</a>
%H A171367 P. Flajolet, <a href="http://algo.inria.fr/flajolet/Publications/Flajolet80b.pdf">Combinatorial aspects of continued fractions</a>, Discrete Mathematics, Volume 32, Issue 2, 1980, pp. 125-161.
%F A171367 G.f.: 1/(1-x^2/(1-x/(1-x^2/(1-2x/(1-x^2/1-3x/(1-x^2/(1-4x/(1-x^2/(1-5x/(1-... (continued fraction).
%F A171367 G.f.: (G(0) - 1)/(x-1)/x where G(k) =  1 - x/(1-k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 16 2013
%F A171367 G.f.: T(0)/(1-x^2), where T(k) = 1-x^3*(k+1)/(x^3*(k+1)-(1-x*(x+k))*(1-x*(x+1+k))/T(k+1) ); (continued fraction, after P. Flajolet, p. 140). - _Sergei N. Gladkovskii_, Oct 30 2013
%F A171367 G.f. (alternating signs): Sum_{k>=0} S(x,k)*x^k, where S(x,k)*exp(-x) is the inverse Mellin transform of Gamma(s)*s^k. - _Benedict W. J. Irwin_, Oct 14 2016
%p A171367 b:= proc(n, m) option remember; `if`(n<=m,
%p A171367      `if`(n=m, 1, 0), m*b(n-1, m)+b(n-1, m+1))
%p A171367     end:
%p A171367 a:= n-> b(n, 0):
%p A171367 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 16 2023
%t A171367 Table[Sum[StirlingS2[n-k, k], {k, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, Oct 18 2016 *)
%o A171367 (Maxima) makelist(sum(stirling2(n-k,k),k,0,n),n,0,60); /* _Emanuele Munarini_, Jun 01 2012 */
%o A171367 (PARI) a(n) = sum(k=0, n, stirling(n-k, k,2)); /* _Joerg Arndt_, Jan 16 2013 */
%Y A171367 Cf. A024427, A097341, A097342.
%K A171367 easy,nonn
%O A171367 0,5
%A A171367 _Paul Barry_, Dec 06 2009