A113350 Triangle Q, read by rows, such that Q^2 transforms column k of Q^2 into column k+1 of Q^2, so that column k of Q^2 equals column 0 of Q^(2*k+2), where Q^2 denotes the matrix square of Q.

1, 2, 1, 5, 4, 1, 19, 22, 6, 1, 113, 166, 51, 8, 1, 966, 1671, 561, 92, 10, 1, 10958, 21510, 7726, 1324, 145, 12, 1, 156700, 341463, 129406, 23010, 2575, 210, 14, 1, 2727794, 6496923, 2572892, 471724, 53935, 4434, 287, 16, 1

Offset

0,2

Links

Table of n, a(n) for n=0..44.

Formula

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

so that triangular matrix Q may be defined by

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

the dual triangular matrix P = A113340 is defined by

[P]_k = [P^(2*k+1)]_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.

Example

Triangle Q begins:
1;
2,1;
5,4,1;
19,22,6,1;
113,166,51,8,1;
966,1671,561,92,10,1;
10958,21510,7726,1324,145,12,1;
156700,341463,129406,23010,2575,210,14,1;
2727794,6496923,2572892,471724,53935,4434,287,16,1;
56306696,144856710,59525136,11198006,1305070,108593,7021,376,18,1;
Matrix square Q^2 begins:
1;
4,1;
18,8,1;
112,68,12,1;
965,712,150,16,1;
10957,9270,2184,264,20,1; ...
where Q^2 transforms column k of Q^2 into column k+1:
at k=0, [Q^2]*[1,4,18,112,965,...] = [1,8,68,712,9270,...];
at k=1, [Q^2]*[1,8,68,712,9270,...] =
[1,12,150,2184,37523,...].

Prog

(PARI) Q(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^(2*k+2))[n-k+1, 1]

Crossrefs

Cf. A113351 (column 1), A113352 (column 2), A113353 (column 3), A113354 (column 4); A113355 (Q^2), A113365 (Q^3), A113340 (P), A113345 (P^2), A113360 (P^3).

Sequence in context: A073107 A248669 A103718 * A227372 A359131 A164678

Adjacent sequences: A113347 A113348 A113349 * A113351 A113352 A113353

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 08 2005

Status

approved