A049378 Row sums of triangle A049353.

1, 1, 6, 46, 436, 4956, 65776, 996976, 16957536, 319259296, 6581662336, 147290942976, 3552885191296, 91827536814976, 2530228890080256, 74003737259670016, 2288810287491774976, 74607500831801289216, 2555587654482227055616, 91746983502042106018816

Offset: 0

Author

Wolfdieter Lang

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..421
• W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Crossrefs

Column k=5 of A291709.

Cf. A049353, A049426.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+3)!/4!*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 + 3)!/4!*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*(2-x)*(2-2*x+x^2)/(4*(1-x)^4) (E.g.f. first column of A049353).

From Seiichi Manyama, Jan 18 2025: (Start)

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004213(k).

a(n) = (1/exp(1/4)) * (-1)^n * n! * Sum_{k>=0} binomial(-4*k,n)/(4^k * k!). (End)

Extensions

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