0,5

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).

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,...].

(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]}

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

Sequence in context: A013561 A111473 A067402 this_sequence A134523 A098778 A078122

Adjacent sequences: A113337 A113338 A113339 this_sequence A113341 A113342 A113343

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 08 2005