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A106208 Triangular matrix T, read by rows, that satisfies: [T^-1](n,k) = -(k+1)*T(n-1,k) when (n-1)>=k>=0, with T(n,n) = 1 and T(n+1,n) = (n+1) for n>=0.

%I A106208 #7 Jun 14 2015 00:51:45
%S A106208 1,1,1,3,2,1,16,10,3,1,127,78,21,4,1,1363,832,216,36,5,1,18628,11342,
%T A106208 2901,460,55,6,1,311250,189286,48081,7456,840,78,7,1,6173791,3752320,
%U A106208 949800,145660,15955,1386,105,8,1,142190703,86392756,21826470,3327340
%N A106208 Triangular matrix T, read by rows, that satisfies: [T^-1](n,k) = -(k+1)*T(n-1,k) when (n-1)>=k>=0, with T(n,n) = 1 and T(n+1,n) = (n+1) for n>=0.
%C A106208 Column 0 is A082161 (offset 1). Column 1 is (1/2)*A102087. Row sums form A106209.
%F A106208 T(n, k) = A102086(n, k)/(k+1) for n>=0, k>=0. T(n, 0) = A082161(n) for n>0, with T(0, 0) = 1. G.f. for column k: 1 = Sum_{n>=0} T(n+k, k)*x^n*prod_{j=1, n+1} (1-(j+k)*x).
%e A106208 Triangle T begins:
%e A106208 1;
%e A106208 1,1;
%e A106208 3,2,1;
%e A106208 16,10,3,1;
%e A106208 127,78,21,4,1;
%e A106208 1363,832,216,36,5,1;
%e A106208 18628,11342,2901,460,55,6,1;
%e A106208 311250,189286,48081,7456,840,78,7,1;
%e A106208 6173791,3752320,949800,145660,15955,1386,105,8,1; ...
%e A106208 Matrix inverse T^-1 begins:
%e A106208 1;
%e A106208 -1,1;
%e A106208 -1,-2,1;
%e A106208 -3,-4,-3,1;
%e A106208 -16,-20,-9,-4,1;
%e A106208 -127,-156,-63,-16,-5,1;
%e A106208 -1363,-1664,-648,-144,-25,-6,1;
%e A106208 -18628,-22684,-8703,-1840,-275,-36,-7,1; ...
%e A106208 where [T^-1](n,k) = -(k+1)*T(n-1,k) when (n-1)>=k>=0.
%e A106208 G.f. for column 0: 1 = 1(1-x) + 1*x*(1-x)(1-2x) +
%e A106208 3*x^2*(1-x)(1-2x)(1-3x) + ... +
%e A106208 T(n,0)*x^n*(1-x)(1-2x)(1-3x)*..*(1-(n+1)*x) + ...
%e A106208 G.f. for column 1: 1 = 1(1-2x) + 2*x*(1-2x)(1-3x) +
%e A106208 10*x^2*(1-2x)(1-3x)(1-4x) + ... +
%e A106208 T(n+1,1)*x^n*(1-2x)(1-3x)(1-4x)*..*(1-(n+2)*x) + ...
%e A106208 G.f. for column 2: 1 = 1(1-3x) + 3*x*(1-3x)(1-4x) +
%e A106208 21*x^2*(1-3x)(1-4x)(1-5x) + ... +
%e A106208 T(n+2,2)*x^n*(1-3x)(1-4x)(1-5x)*..*(1-(n+3)*x) + ...
%o A106208 (PARI) T(n,k)=if(n<k,0,if(n==k,1,polcoeff( 1-sum(i=0,n-k-1,T(i+k,k)*x^i*prod(j=1,i+1,1-(j+k)*x+x*O(x^(n-k)))),n-k)))
%o A106208 (PARI) T(n,k)=local(A=matrix(1,1),B);A[1,1]=1; for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=j,if(j==1,B[i,j]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]));));A=B);return(A[n+1,k+1]/(k+1))
%Y A106208 Cf. A102086, A082161, A106209.
%K A106208 nonn,tabl
%O A106208 0,4
%A A106208 _Paul D. Hanna_, May 01 2005