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

0,2

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

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

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}]

Prog

(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))}

Crossrefs

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

Cf. A000110.

Sequence in context: A030898 A002788 A332796 * A009575 A263687 A180891

Adjacent sequences: A134091 A134092 A134093 * A134095 A134096 A134097

Keyword

nonn

Author

Paul D. Hanna, Oct 08 2007

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

Status

approved