0,4

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

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).

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) + ...

(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)))} (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))}

Cf. A102086, A082161, A106211.

Sequence in context: A001497 A123244 A105599 this_sequence A033842 A104417 A121438

Adjacent sequences: A106207 A106208 A106209 this_sequence A106211 A106212 A106213

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), May 01 2005