OFFSET
0,3
FORMULA
T(n, 0) = A082161(n) for n>0, with T(0, 0) = 1.
G.f. for column k: T(k, k) = k+1 = Sum_{n>=0} T(n+k, k)*x^n*prod_{j=1, n+1} (1-(j+k)*x).
EXAMPLE
Rows of T begin:
[1],
[1,2],
[3,4,3],
[16,20,9,4],
[127,156,63,16,5],
[1363,1664,648,144,25,6],
[18628,22684,8703,1840,275,36,7],
[311250,378572,144243,29824,4200,468,49,8],
[6173791,7504640,2849400,582640,79775,8316,735,64,9],...
Matrix square T^2 equals T excluding the main diagonal:
[1],
[3,4],
[16,20,9],
[127,156,63,16],
[1363,1664,648,144,25],...
G.f. for column 0: 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: 2 = 2(1-2x) + 4*x*(1-2x)(1-3x) + 20*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: 3 = 3(1-3x) + 9*x*(1-3x)(1-4x) + 63*x^2*(1-3x)(1-4x)(1-5x) + ... + T(n+2,2)*x^n*(1-3x)(1-4x)(1-5x)*..*(1-(n+3)*x) + ...
MAPLE
{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])}
MATHEMATICA
T[n_, n_] := n+1; T[n_, k_] /; k>n = 0; T[n_, k_] /; k == n-1 := n^2; T[n_, k_] := T[n, k] = Coefficient[1-Sum[T[i, k]*x^i*Product[1-(j+k)*x, {j, 1, i-k+1}], {i, k, n-1}], x, n]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 15 2014, after PARI script *)
PROG
(PARI) {T(n, k)=if(n<k, 0, if(n==k, k+1, polcoeff(1-sum(i=k, n-1, T(i, k)*x^i*prod(j=1, i-k+1, 1-(j+k)*x+x*O(x^n))), n)))}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 29 2004
STATUS
approved