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

%I A134094 #12 Oct 17 2019 11:57:33
%S A134094 1,2,6,26,140,887,6405,51564,455712,4370567,45081476,496556194,
%T A134094 5806502663,71734434956,932447207866,12707973761320,181033752071568,
%U A134094 2688530124711819,41525910256013832,665674913113633582
%N A134094 Binomial convolution of the Stirling numbers of the second kind.
%C A134094 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.
%H A134094 Robert Israel, <a href="/A134094/b134094.txt">Table of n, a(n) for n = 0..517</a>
%F A134094 a(n) = sum( C(n+1,k)*|S2(n,k)|, k=0..n).
%F A134094 Row sums of triangle A134090.
%F A134094 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.
%F A134094 a(n) = Sum_{k=0..n} C(n+1,k)*S2(n,k). From _Olivier Gérard_, Oct 23 2012
%p A134094 f:= proc(n) local k; add(binomial(n+1,k)*combinat:-stirling2(n,k),k=0..n) end proc:
%p A134094 map(f, [$0..30]); # _Robert Israel_, Oct 16 2019
%t A134094 Table[Sum[Binomial[n + 1, k] StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
%o A134094 (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))}
%Y A134094 Cf. A134090; columns: A122455, A134091, A134092, A134093; A048993 (S2).
%Y A134094 Cf. A000110.
%K A134094 nonn
%O A134094 0,2
%A A134094 _Paul D. Hanna_, Oct 08 2007
%E A134094 Definition modified and Mathematica program by _Olivier Gérard_, Oct 23 2012
%E A134094 Simplified Name and moved formulas into the formula section. - _Paul D. Hanna_, Oct 23 2013