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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. 32
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Table of n, a(n) for n=0..54.

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).

EXAMPLE

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,...].

PROG

(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]

CROSSREFS

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

Sequence in context: A111473 A234944 A067402 * A134523 A098778 A078122

Adjacent sequences: A113337 A113338 A113339 * A113341 A113342 A113343

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Nov 08 2005

STATUS

approved

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Last modified February 5 20:57 EST 2023. Contains 360087 sequences. (Running on oeis4.)