A113350 - OEIS

%I #5 Jun 13 2017 22:51:36

%S 1,2,1,5,4,1,19,22,6,1,113,166,51,8,1,966,1671,561,92,10,1,10958,

%T 21510,7726,1324,145,12,1,156700,341463,129406,23010,2575,210,14,1,

%U 2727794,6496923,2572892,471724,53935,4434,287,16,1

%N 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.

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

%F so that triangular matrix Q may be defined by

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

%F the dual triangular matrix P = A113340 is defined by

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

%F Then, amazingly, powers of P and Q satisfy:

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

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

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

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

%F Also, we have the column transformations:

%F P^2 * [P]_k = [P]_{k+1},

%F P^2 * [Q]_k = [Q]_{k+1},

%F Q^2 * [P^2]_k = [P^2]_{k+1},

%F Q^2 * [Q^2]_k = [Q^2]_{k+1},

%F for all k>=0.

%e Triangle Q begins:

%e 1;

%e 2,1;

%e 5,4,1;

%e 19,22,6,1;

%e 113,166,51,8,1;

%e 966,1671,561,92,10,1;

%e 10958,21510,7726,1324,145,12,1;

%e 156700,341463,129406,23010,2575,210,14,1;

%e 2727794,6496923,2572892,471724,53935,4434,287,16,1;

%e 56306696,144856710,59525136,11198006,1305070,108593,7021,376,18,1;

%e Matrix square Q^2 begins:

%e 1;

%e 4,1;

%e 18,8,1;

%e 112,68,12,1;

%e 965,712,150,16,1;

%e 10957,9270,2184,264,20,1; ...

%e where Q^2 transforms column k of Q^2 into column k+1:

%e at k=0, [Q^2]*[1,4,18,112,965,...] = [1,8,68,712,9270,...];

%e at k=1, [Q^2]*[1,8,68,712,9270,...] =

%e [1,12,150,2184,37523,...].

%o (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]

%Y 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).

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Nov 08 2005