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.

1, 1, 1, 3, 3, 1, 16, 16, 5, 1, 127, 127, 39, 7, 1, 1363, 1363, 416, 72, 9, 1, 18628, 18628, 5671, 967, 115, 11, 1, 311250, 311250, 94643, 16027, 1864, 168, 13, 1, 6173791, 6173791, 1876160, 316600, 36415, 3191, 231, 15, 1, 142190703, 142190703

Offset: 0

Author

Paul D. Hanna, May 01 2005

Keywords

• nonn
• tabl

Comments

Both column 0 and column 1 form A082161. Row sums form A106211.

Examples

			Triangle T begins:
1;
1,1;
3,3,1;
16,16,5,1;
127,127,39,7,1;
1363,1363,416,72,9,1;
18628,18628,5671,967,115,11,1;
311250,311250,94643,16027,1864,168,13,1;
6173791,6173791,1876160,316600,36415,3191,231,15,1; ...
Matrix inverse T^-1 begins:
1;
-1,1;
0,-3,1;
0,-1,-5,1;
0,-3,-4,-7,1;
0,-16,-20,-9,-9,1;
0,-127,-156,-63,-16,-11,1;
0,-1363,-1664,-648,-144,-25,-13,1;
0,-18628,-22684,-8703,-1840,-275,-36,-15,1; ...
where [T^-1](n,k) = -k^2*T(n-2,k) when (n-2)>=k>=0.
G.f. for column 0: 1/(1-0x) = 1*(1-1x) + 1*x*(1-1x)(1-2x) +
3*x^2*(1-1x)(1-2x)(1-3x) + 16*x^3*(1-1x)(1-2x)(1-3x)(1-4x) + ...
+ T(n,0)*x^n*(1-1x)(1-2x)*..*(1-(n+1)*x) + ...
G.f. for column 1: 1/(1-1x) = 1*(1-2x) + 3*x*(1-2x)(1-3x) +
16*x^2*(1-2x)(1-3x)(1-4x) + 127*x^3*(1-2x)(1-3x)(1-4x)(1-5x) + ...
+ T(n+1,1)*x^n*(1-2x)(1-3x)*..*(1-(n+2)*x) + ...
G.f. for column 2: 1/(1-2x) = 1*(1-3x) + 5*x*(1-3x)(1-4x) +
39*x^2*(1-3x)(1-4x)(1-5x) + 416*x^3*(1-3x)(1-4x)(1-5x)(1-6x) + ...
+ T(n+2,2)*x^n*(1-3x)(1-4x)*..*(1-(n+3)*x) + ...
		

Crossrefs

Cf. A102086, A082161, A106211.

Programs

PARI

T(n,k)=if(n
				

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))

Formula

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).