_Paul D. Hanna (pauldhanna(AT)juno.com), _, Oct 14 2005

Let GF[T] denote the g.f. of triangular matrix T. Then GF[T] = 1 + x*(1+y)*GF[T^3], and for all integer p>=1: GF[T^p] = 1 + x*Sum_{j=1..p} GF[T^(p+2*j)] + x*y*GF[T^(3*p)].

nonn,tabl,new

nonn,tabl,new

Paul D . Hanna (pauldhanna(AT)juno.com), Oct 14 2005

Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^3](n-1,k-1) + [T^3](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^3 is the matrix 3-rd third power of T.

nonn,tabl,new

Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^3](n-1,k-1) + [T^3](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^3 is the matrix 3-rd power of T.

1, 1, 1, 3, 4, 1, 21, 33, 13, 1, 331, 586, 294, 40, 1, 11973, 23299, 13768, 2562, 121, 1, 1030091, 2166800, 1447573, 333070, 22569, 364, 1, 218626341, 490872957, 361327779, 97348117, 8466793, 200931, 1093, 1, 118038692523, 280082001078

0,4

Column 0 of the matrix power p, T^p, equals the number of 3-tournament sequences having initial term p.

Let GF[T] denote the g.f. of triangular matrix T. Then GF[T] = 1 + x*(1+y)*GF[T^3], and for all integer p>=1: GF[T^p] = 1 + x*Sum_{j=1..p} GF[T^(p+2*j)] + x*y*GF[T^(3*p)].

Triangle T begins:

1;

1,1;

3,4,1;

21,33,13,1;

331,586,294,40,1;

11973,23299,13768,2562,121,1;

1030091,2166800,1447573,333070,22569,364,1; ...

Matrix square T^2 (A113088) begins:

1;

2,1;

10,8,1;

114,118,26,1;

2970,3668,1108,80,1;

182402,257122,96416,9964,242,1; ...

where column 0 equals A113089.

Matrix cube T^3 (A113090) begins:

1;

3,1;

21,12,1;

331,255,39,1;

11973,11326,2442,120,1;

1030091,1136709,310864,22206,363,1; ...

where adjacent sums in row n of T^3 forms row n+1 of T.

(PARI) {T(n, k)=local(M=matrix(n+1, n+1)); for(r=1, n+1, for(c=1, r, M[r, c]=if(r==c, 1, if(c>1, (M^3)[r-1, c-1])+(M^3)[r-1, c]))); return(M[n+1, k+1])}

Cf. A113081; A097710, A113095, A113106; A113085 (column 0), A113088 (T^2), A113087 (row sums).

nonn,tabl

Paul D Hanna (pauldhanna(AT)juno.com), Oct 14 2005

approved