A113090 -id:A113090 - cp's OEIS frontend

A113091 Column 1 of triangle A113090.

1, 12, 255, 11326, 1136709, 272246616, 162043308555, 246976754748242, 985451332639242321, 10465989453804481920460, 299684548041297413533699479, 23373481327888949572209959486022, 5007060652963472122780384849366677469

Offset: 1

Paul D. Hanna, Oct 14 2005

nonn

Cf. A113090.

{a(n)=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(if(n<1,0,(M^3)[n+1,2]))}

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.

Original entry on oeis.org

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

Paul D. Hanna, Oct 14 2005

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

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

{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])}

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)].

cp's OEIS Frontend

A113091 Column 1 of triangle A113090.

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