A113084 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 third 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

Offset: 0

Author

Paul D. Hanna, Oct 14 2005

Keywords

• nonn
• tabl

Comments

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

Examples

			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.
		

Crossrefs

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

Programs

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])}

Formula

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)].