A113340 Triangle P, read by rows, such that P^2 transforms column k of P into column k+1 of P, so that column k of P equals column 0 of P^(2*k+1), where P^2 denotes the matrix square of P.

1, 1, 1, 1, 3, 1, 1, 12, 5, 1, 1, 69, 35, 7, 1, 1, 560, 325, 70, 9, 1, 1, 6059, 3880, 889, 117, 11, 1, 1, 83215, 57560, 13853, 1881, 176, 13, 1, 1, 1399161, 1030751, 258146, 36051, 3421, 247, 15, 1, 1, 28020221, 21763632, 5633264, 805875, 77726, 5629, 330, 17, 1

Offset: 0

Author

Paul D. Hanna, Nov 08 2005

Keywords

• nonn
• tabl

Examples

			Triangle P begins:
1;
1,1;
1,3,1;
1,12,5,1;
1,69,35,7,1;
1,560,325,70,9,1;
1,6059,3880,889,117,11,1;
1,83215,57560,13853,1881,176,13,1;
1,1399161,1030751,258146,36051,3421,247,15,1;
1,28020221,21763632,5633264,805875,77726,5629,330,17,1;
1,654110586,531604250,141487178,20661609,2023461,147810,8625,425,19,1;
Matrix square P^2 (A113345) starts:
1;
2,1;
5,6,1;
19,39,10,1;
113,327,105,14,1;
966,3556,1315,203,18,1; ...
where P^2 transforms column k of P into column k+1 of P:
at k=0, [P^2]*[1,1,1,1,1,...] = [1,3,12,69,560,...];
at k=1, [P^2]*[1,3,12,69,560,...] = [1,5,35,325,3880,...].
		

Crossrefs

Cf. A113341 (column 1), A113342 (column 2), A113343 (column 3), A113344 (column 4); A113345 (P^2), A113360 (P^3), A113350 (Q).

Programs

PARI

P(n,k)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B);A[n+1,k+1]

Formula

Let [P^m]_k denote column k of matrix power P^m,

so that triangular matrix P may be defined by

[P]_k = [P^(2*k+1)]_0, for k>=0.

Define the dual triangular matrix Q = A113350 by

[Q]_k = [P^(2*k+2)]_0, for k>=0.

Then, amazingly, powers of P and Q satisfy:

[P^(2*j+1)]_k = [P^(2*k+1)]_j,

[P^(2*j+2)]_k = [Q^(2*k+1)]_j,

[Q^(2*j+2)]_k = [Q^(2*k+2)]_j,

for all j>=0, k>=0.

Also, we have the column transformations:

P^2 * [P]k = [P]{k+1},

P^2 * [Q]k = [Q]{k+1},

Q^2 * [P^2]k = [P^2]{k+1},

Q^2 * [Q^2]k = [Q^2]{k+1},

for all k>=0.

Further, g.f.s of P and Q satisfy:

GF(P) = 1/(1-x) + x*y*GF(Q^2*P^-1),

GF(Q^-1*P^2) = 1 + x*y*GF(Q).