A134094 Binomial convolution of the Stirling numbers of the second kind.

1, 2, 6, 26, 140, 887, 6405, 51564, 455712, 4370567, 45081476, 496556194, 5806502663, 71734434956, 932447207866, 12707973761320, 181033752071568, 2688530124711819, 41525910256013832, 665674913113633582

Offset: 0

Author

Paul D. Hanna, Oct 08 2007

Keywords

• nonn

Comments

Row n of triangle T=A134090 = row n of (I + D*C)^n for n>=0 where C denotes Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere.

Links

• Robert Israel, Table of n, a(n) for n = 0..517

Crossrefs

Cf. A134090; columns: A122455, A134091, A134092, A134093; A048993 (S2).

Cf. A000110.

Programs

Maple

f:= proc(n) local k; add(binomial(n+1,k)*combinat:-stirling2(n,k),k=0..n) end proc:
map(f, [$0..30]); # Robert Israel, Oct 16 2019

Mathematica

Table[Sum[Binomial[n + 1, k] StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]

PARI

{a(n)=sum(k=0,n,binomial(n,k)*polcoeff((1-k*x)/prod(i=0,k+1,1-i*x+x*O(x^(n))),n-k))}

Formula

a(n) = sum( C(n+1,k)*|S2(n,k)|, k=0..n).

Row sums of triangle A134090.

a(n) = [x^n] Sum_{k=0..n} C(n,k)*x^k*(1-k*x) / [Product_{i=0..k+1}(1-i*x)], equivalently, a(n) = Sum_{k=0..n} C(n,k)*[S2(n,k) - k*S2(n-1,k)], where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.

a(n) = Sum_{k=0..n} C(n+1,k)*S2(n,k). From Olivier Gérard, Oct 23 2012

Extensions

Definition modified and Mathematica program by Olivier Gérard, Oct 23 2012

Simplified Name and moved formulas into the formula section. - Paul D. Hanna, Oct 23 2013