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A134094 Binomial convolution of the Stirling numbers of the second kind. 8
1, 2, 6, 26, 140, 887, 6405, 51564, 455712, 4370567, 45081476, 496556194, 5806502663, 71734434956, 932447207866, 12707973761320, 181033752071568, 2688530124711819, 41525910256013832, 665674913113633582 (list; graph; refs; listen; history; text; internal format)
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
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
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

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Last modified July 21 22:18 EDT 2024. Contains 374477 sequences. (Running on oeis4.)