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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 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

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