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.

1, 1, 1, 3, 2, 1, 16, 10, 3, 1, 127, 78, 21, 4, 1, 1363, 832, 216, 36, 5, 1, 18628, 11342, 2901, 460, 55, 6, 1, 311250, 189286, 48081, 7456, 840, 78, 7, 1, 6173791, 3752320, 949800, 145660, 15955, 1386, 105, 8, 1, 142190703, 86392756, 21826470, 3327340

Offset: 0

Author

Paul D. Hanna, May 01 2005

Keywords

• nonn
• tabl

Comments

Column 0 is A082161 (offset 1). Column 1 is (1/2)*A102087. Row sums form A106209.

Examples

			Triangle T begins:
1;
1,1;
3,2,1;
16,10,3,1;
127,78,21,4,1;
1363,832,216,36,5,1;
18628,11342,2901,460,55,6,1;
311250,189286,48081,7456,840,78,7,1;
6173791,3752320,949800,145660,15955,1386,105,8,1; ...
Matrix inverse T^-1 begins:
1;
-1,1;
-1,-2,1;
-3,-4,-3,1;
-16,-20,-9,-4,1;
-127,-156,-63,-16,-5,1;
-1363,-1664,-648,-144,-25,-6,1;
-18628,-22684,-8703,-1840,-275,-36,-7,1; ...
where [T^-1](n,k) = -(k+1)*T(n-1,k) when (n-1)>=k>=0.
G.f. for column 0: 1 = 1(1-x) + 1*x*(1-x)(1-2x) +
3*x^2*(1-x)(1-2x)(1-3x) + ... +
T(n,0)*x^n*(1-x)(1-2x)(1-3x)*..*(1-(n+1)*x) + ...
G.f. for column 1: 1 = 1(1-2x) + 2*x*(1-2x)(1-3x) +
10*x^2*(1-2x)(1-3x)(1-4x) + ... +
T(n+1,1)*x^n*(1-2x)(1-3x)(1-4x)*..*(1-(n+2)*x) + ...
G.f. for column 2: 1 = 1(1-3x) + 3*x*(1-3x)(1-4x) +
21*x^2*(1-3x)(1-4x)(1-5x) + ... +
T(n+2,2)*x^n*(1-3x)(1-4x)(1-5x)*..*(1-(n+3)*x) + ...
		

Crossrefs

Cf. A102086, A082161, A106209.

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+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))

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 = Sum_{n>=0} T(n+k, k)*x^n*prod_{j=1, n+1} (1-(j+k)*x).