A049377 Row sums of triangle A049352.

1, 1, 5, 33, 273, 2721, 31701, 421905, 6302913, 104270913, 1889862021, 37204038081, 789866524305, 17977594555233, 436435929785493, 11251798888929201, 306889765901872641, 8825681949708120705, 266828094135981378693, 8458295877281844310113

Offset: 0

Author

Wolfdieter Lang

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..428
• W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Crossrefs

Column k=4 of A291709.

Cf. A049352, A049425.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+2)!/6*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017

Mathematica

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*(j+2)!/6*a[n-j], {j, 1, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Formula

E.g.f. exp(p(x)) with p(x) := x*(3-3*x+x^2)/(3*(1-x)^3) (E.g.f. first column of A049352).

a(n) ~ n^(n-1/8)/2 * exp(-1/4 + 5*n^(1/4)/24 + sqrt(n)/2 + 4*n^(3/4)/3 - n). - Vaclav Kotesovec, Oct 23 2017

E.g.f.: Sum_{n>=0} ( Integral 1/(1-x)^4 dx )^n / n!, where the constant of integration is taken to be zero. - Paul D. Hanna, Apr 27 2019

From Seiichi Manyama, Jan 18 2025: (Start)

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004212(k).

a(n) = (1/exp(1/3)) * (-1)^n * n! * Sum_{k>=0} binomial(-3*k,n)/(3^k * k!). (End)

Extensions

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017