T(n,k) tabl head (triangle) for A094816 (a Sheffer matrix) n\k 0 1 2 3 4 5 6 7 8 9 ... 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 2 1 3 1 0 0 0 0 0 0 0 3 1 8 6 1 0 0 0 0 0 0 4 1 24 29 10 1 0 0 0 0 0 5 1 89 145 75 15 1 0 0 0 0 6 1 415 814 545 160 21 1 0 0 0 7 1 2372 5243 4179 1575 301 28 1 0 0 8 1 16072 38618 34860 15659 3836 518 36 1 0 9 1 125673 321690 318926 163191 47775 8274 834 45 1 . . . E.g.f. of column k sequences: exp(x)*((-ln(1-x))^k)/k!, k>=0. This is an example for a Sheffer matrix (binomial convolution matrix). Row sums give: [1, 2, 5, 16, 65, 326, 1957, 13700, 109601, 986410, ...] = A000522. e.g.f.: exp(x)/(1-x). Alternating row sums give 1-n= A024000(n): [1, 0, -1, -2, -3, -4, -5, -6, -7, -8,...] e.g.f.: exp(x)*(1-x). #################### For the z- and a-sequence of for a Sheffer matrix see the W. Lang link under A006232. In the case at hand S=(exp(x),-ln(1-x)) (our notation) these sequences are: a(n)=B(n)= A027641(n)/A027642(n) (Bernoulli numbers) with e.g.f. x/(exp(x)-1), z(n)= A130189(n)/ A130190(n) with e.g.f. h(x)*(1-exp(h(x))) with h(x):=1-exp(-x). This leads to the following recurrences for T(n,k)= A094816(n,m): T(n,0) = n*sum(z(n)*T(n-1,k),k=0..n-1), n>=1, with T(0,0)=1 as input. T(n,k) = (n/k)*sum(binomial(k-1+j,m-1)*a(j)*T(n-1,k-1+j), n>=m>=1. a(n)=B(n)=[1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, 0, 43867/798, 0, -174611/330, 0, 854513/138, 0, -236364091/2730, 0, 8553103/6, 0, -23749461029/870, 0, 8615841276005/14322, 0, -7709321041217/510, 0, 2577687858367/6, 0, -26315271553053477373/1919190, 0, 2929993913841559/6, 0,...] z(n)= [1, -1/2, 5/6, -7/4, 68/15, -167/12, 2057/42, -4637/24, 75703/90, -39941/10, 676360/33, -902547/8, 602501827/910, -432761746/105, 2438757091/90, -8997865117/48, 346824403906/255, -1857709421899/180, 325976550837563/3990, -282728710837871/420] ##################### A relation for the row polynomials c(n,x):=sum(T(n,k)*x^k,k=0..n) (Poisson-Charlier case) is (1-exp(-d_x))*c(n,x) = n*c(n-1,x), n>=1, with d_x the differentiation operator, hence c(n,x) - c(n,x-1) = n*c(n-1,x), n>=1. This is an application of Theorem Th. 2.3.7. in S. Roman's book (see the W. Lang link under A006232), p.20. See also p.121 of this book, eq. 4.3.9. A recurrence is c(n,x) = c(n-1,x)+x*c(n-1,x+1), n>=1. This is an application of Corollary 3.7.2., p.50, in S. Roman's book. See also p.121. ####################################### e.o.f. ##################################################