cp's OEIS Frontend

A106210 Triangular matrix T, read by rows, that satisfies: [T^-1](n,k) = -k^2*T(n-2,k) when (n-2)>=k>=0, with T(n,n) = 1 and T(n+1,n) = (2*n+1) for n>=0.

%I A106210 #7 Jun 14 2015 00:51:15
%S A106210 1,1,1,3,3,1,16,16,5,1,127,127,39,7,1,1363,1363,416,72,9,1,18628,
%T A106210 18628,5671,967,115,11,1,311250,311250,94643,16027,1864,168,13,1,
%U A106210 6173791,6173791,1876160,316600,36415,3191,231,15,1,142190703,142190703
%N A106210 Triangular matrix T, read by rows, that satisfies: [T^-1](n,k) = -k^2*T(n-2,k) when (n-2)>=k>=0, with T(n,n) = 1 and T(n+1,n) = (2*n+1) for n>=0.
%C A106210 Both column 0 and column 1 form A082161. Row sums form A106211.
%F A106210 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/(1-k*x) = Sum_{n>=0} T(n+k, k)*x^n*prod_{j=1, n+1} (1-(j+k)*x).
%e A106210 Triangle T begins:
%e A106210 1;
%e A106210 1,1;
%e A106210 3,3,1;
%e A106210 16,16,5,1;
%e A106210 127,127,39,7,1;
%e A106210 1363,1363,416,72,9,1;
%e A106210 18628,18628,5671,967,115,11,1;
%e A106210 311250,311250,94643,16027,1864,168,13,1;
%e A106210 6173791,6173791,1876160,316600,36415,3191,231,15,1; ...
%e A106210 Matrix inverse T^-1 begins:
%e A106210 1;
%e A106210 -1,1;
%e A106210 0,-3,1;
%e A106210 0,-1,-5,1;
%e A106210 0,-3,-4,-7,1;
%e A106210 0,-16,-20,-9,-9,1;
%e A106210 0,-127,-156,-63,-16,-11,1;
%e A106210 0,-1363,-1664,-648,-144,-25,-13,1;
%e A106210 0,-18628,-22684,-8703,-1840,-275,-36,-15,1; ...
%e A106210 where [T^-1](n,k) = -k^2*T(n-2,k) when (n-2)>=k>=0.
%e A106210 G.f. for column 0: 1/(1-0x) = 1*(1-1x) + 1*x*(1-1x)(1-2x) +
%e A106210 3*x^2*(1-1x)(1-2x)(1-3x) + 16*x^3*(1-1x)(1-2x)(1-3x)(1-4x) + ...
%e A106210 + T(n,0)*x^n*(1-1x)(1-2x)*..*(1-(n+1)*x) + ...
%e A106210 G.f. for column 1: 1/(1-1x) = 1*(1-2x) + 3*x*(1-2x)(1-3x) +
%e A106210 16*x^2*(1-2x)(1-3x)(1-4x) + 127*x^3*(1-2x)(1-3x)(1-4x)(1-5x) + ...
%e A106210 + T(n+1,1)*x^n*(1-2x)(1-3x)*..*(1-(n+2)*x) + ...
%e A106210 G.f. for column 2: 1/(1-2x) = 1*(1-3x) + 5*x*(1-3x)(1-4x) +
%e A106210 39*x^2*(1-3x)(1-4x)(1-5x) + 416*x^3*(1-3x)(1-4x)(1-5x)(1-6x) + ...
%e A106210 + T(n+2,2)*x^n*(1-3x)(1-4x)*..*(1-(n+3)*x) + ...
%o A106210 (PARI) T(n,k)=if(n<k,0,if(n==k,1,polcoeff( 1/(1-k*x)-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 A106210 (PARI) T(n,k)=local(A=matrix(1,1),B);A[1,1]=1; for(m=2,n+2,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(if(k==0,if(n==0,1,A[n+1,k+1]),A[n+1,k]/k^2))
%Y A106210 Cf. A102086, A082161, A106211.
%K A106210 nonn,tabl
%O A106210 0,4
%A A106210 _Paul D. Hanna_, May 01 2005