A102086 Triangular matrix, read by rows, that satisfies: T(n,k) = [T^2](n-1,k) when n>k>=0, with T(n,n) = (n+1).
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, 142190703, 172785512
Offset: 0
Examples
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) + ...
Programs
-
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 *) -
PARI
{T(n,k)=if(n
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).
Comments