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%I #6 Mar 31 2012 13:19:59
%S 1,1,1,4,10,1,30,132,27,1,336,2232,696,52,1,5040,46320,19500,2200,85,
%T 1,95040,1141920,606960,91800,5340,126,1,2162160,32639040,20991600,
%U 3986640,310170,11004,175,1,57657600,1061746560,802287360,183550080
%N Triangle of coefficients of certain Sheffer-polynomials.
%C s(n,x) := sum(a(n,m)*x^m,m=0..n) are monic polynomials satisfying s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials p(n,x)=sum(A048786(n,m)*x^m, m=1..n) (row polynomials of triangle A048786) and p(0,x)=1. In the umbral calculus (see reference) the s(n,x) are called Sheffer polynomials for(c(t/(1+4*t)),t/(1+4*t)), where c(x) = g.f. for Catalan numbers A000108. a(n,0) = A001761(n-2) = n!*A000108(n).
%D S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
%F a(n, m) = (n!/m!)*A046527(n, m) = (n!/m!)*binomial(n, m-1)*(4^(n-m+1)-binomial(2*n, n)/binomial(2*(m-1), m-1))/2, n >= m >= 0, a(n, m) := 0, n<m.
%Y A046527, A048786, A000108, A001761.
%K easy,nonn,tabl
%O 0,4
%A _Wolfdieter Lang_
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