0,2

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.

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,...].

(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]}

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: A110271 A073107 A103718 this_sequence A164678 A164679 A061579

Adjacent sequences: A113347 A113348 A113349 this_sequence A113351 A113352 A113353

nonn,tabl

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