A049428 Row sums of triangle A049411.

1, 1, 6, 36, 246, 2046, 19716, 209616, 2441916, 31050396, 425883816, 6244077456, 97391939976, 1609040166696, 28029696862896, 512903202039936, 9829166157390096, 196739739722616336, 4102788435212513376, 88945209649582514496, 2000700796384204930656

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..485
• Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
• W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Crossrefs

Column k=5 of A293991.

Cf. A005012.

Programs

Mathematica

nmax = 20;
a[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, nmax}]];
a[0] = 1; a[n_] := Sum[a[n, m], {m, 1, n}];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 27 2018 *)

Formula

E.g.f.: exp((-1+(1+x)^6)/6).

a(n) = n! * Sum_{k=1..n} Sum_{j=0..k} binomial(6*j,n) *(-1)^(k-j)/ (6^k*(k-j)!*j!). - Vladimir Kruchinin, Feb 07 2011

D-finite with recurrence a(n) -a(n-1) +5*(-n+1)*a(n-2) -10*(n-1)*(n-2)*a(n-3) -10*(n-1)*(n-2)*(n-3)*a(n-4) -5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-6)=0. - R. J. Mathar, Jun 23 2023

a(n) = Sum_{k=0..n} Stirling1(n,k) * A005012(k). - Seiichi Manyama, Jan 31 2024

a(n) = (1/exp(1/6)) * n! * Sum_{k>=0} binomial(6*k,n)/(6^k * k!). - Seiichi Manyama, Jan 18 2025

Extensions

Offset adjusted by R. J. Mathar, Aug 29 2009