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

1, 0, 1, 1, 2, 4, 9, 22, 58, 164, 495, 1587, 5379, 19195, 71872, 281571, 1151338, 4902687, 21696505, 99598840, 473466698, 2327173489, 11810472444, 61808852380, 333170844940, 1847741027555, 10532499571707, 61649191750137, 370208647200165, 2278936037262610, 14369780182166215

Offset

0,5

Links

Seiichi Manyama, Table of n, a(n) for n = 0..665

P. Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics, Volume 32, Issue 2, 1980, pp. 125-161.

Formula

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).

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

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

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

Maple

b:= proc(n, m) option remember; `if`(n<=m,
     `if`(n=m, 1, 0), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2023

Mathematica

Table[Sum[StirlingS2[n-k, k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 18 2016 *)

Prog

(Maxima) makelist(sum(stirling2(n-k, k), k, 0, n), n, 0, 60); [Emanuele Munarini, Jun 01 2012]
(PARI) a(n) = sum(k=0, n, stirling(n-k, k, 2)); /* Joerg Arndt, Jan 16 2013 */

Crossrefs

Cf. A024427, A097341, A097342.

Sequence in context: A249560 A121953 A024427 * A092920 A177377 A321994

Adjacent sequences: A171364 A171365 A171366 * A171368 A171369 A171370

Keyword

easy,nonn

Author

Paul Barry, Dec 06 2009

Status

approved