0,4

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

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).

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) + ...

(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)))} (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))}

Cf. A102086, A082161, A106209.

Sequence in context: A136217 A136220 A088956 this_sequence A129377 A136733 A117269

Adjacent sequences: A106205 A106206 A106207 this_sequence A106209 A106210 A106211

nonn,tabl

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