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%I #6 Jun 13 2017 23:23:07
%S 1,2,1,6,5,1,37,45,8,1,429,635,120,11,1,7629,12815,2556,231,14,1,
%T 185776,343815,71548,6556,378,17,1,5817106,11651427,2508528,233706,
%U 13391,561,20,1,224558216,480718723,106427700,10069521,579047,23817,780,23,1
%N Triangle Q, read by rows, such that Q^3 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^(3*k+2), where Q^3 denotes the matrix cube of Q.
%C Related matrix products are: R^3*Q^-2 (A114154), Q^-2*P^3 (A114155).
%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^(3*k+2)]_0, k>=0,
%F where the triangular matrix P = A113370 satisfies:
%F [P]_k = [P^(3*k+1)]_0, k>=0.
%F Define the triangular matrix R = A113389 by
%F [R]_k = [P^(3*k+3)]_0, k>=0.
%F Then P, Q and R are related by:
%F Q^2 = R*P = R*Q*(R^-2)*Q*R = P*Q*(P^-2)*Q*P,
%F P^2 = Q*(R^-2)*Q^3, R^2 = Q^3*(P^-2)*Q.
%F Amazingly, columns in powers of P, Q, R, obey:
%F [P^(3*j+1)]_k = [P^(3*k+1)]_j,
%F [Q^(3*j+1)]_k = [P^(3*k+2)]_j,
%F [R^(3*j+1)]_k = [P^(3*k+3)]_j,
%F [Q^(3*j+2)]_k = [Q^(3*k+2)]_j,
%F [R^(3*j+2)]_k = [Q^(3*k+3)]_j,
%F [R^(3*j+3)]_k = [R^(3*k+3)]_j,
%F for all j>=0, k>=0.
%F Also, we have the column transformations:
%F P^3 * [P]_k = [P]_{k+1},
%F P^3 * [Q]_k = [Q]_{k+1},
%F P^3 * [R]_k = [R]_{k+1},
%F Q^3 * [P^2]_k = [P^2]_{k+1},
%F Q^3 * [Q^2]_k = [Q^2]_{k+1},
%F Q^3 * [R^2]_k = [R^2]_{k+1},
%F R^3 * [P^3]_k = [P^3]_{k+1},
%F R^3 * [Q^3]_k = [Q^3]_{k+1},
%F R^3 * [R^3]_k = [R^3]_{k+1},
%F for all k>=0.
%e Triangle Q begins:
%e 1;
%e 2,1;
%e 6,5,1;
%e 37,45,8,1;
%e 429,635,120,11,1;
%e 7629,12815,2556,231,14,1;
%e 185776,343815,71548,6556,378,17,1;
%e 5817106,11651427,2508528,233706,13391,561,20,1;
%e 224558216,480718723,106427700,10069521,579047,23817,780,23,1;
%e Matrix square Q^2 (A113384) starts:
%e 1;
%e 4,1;
%e 22,10,1;
%e 212,130,16,1;
%e 3255,2365,328,22,1;
%e 70777,57695,8640,616,28,1; ...
%e Matrix cube Q^3 (A113387) starts:
%e 1;
%e 6,1;
%e 48,15,1;
%e 605,255,24,1;
%e 11196,5630,624,33,1;
%e 280440,159210,19484,1155,42,1; ...
%e where Q^3 transforms column k of Q^2 into column k+1:
%e at k=0, [Q^3]*[1,4,22,212,3255,...] = [1,10,130,2365,...];
%e at k=1, [Q^3]*[1,10,130,2365,...] = [1,16,328,8640,...].
%o (PARI) Q(n,k)=local(A,B);A=Mat(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^(3*j-2))[i-j+1,1]));));A=B);(A^(3*k+2))[n-k+1,1]
%Y Cf. A113375 (column 0), A113382 (column 1), A113383 (column 2).
%Y Cf. A113370 (P), A113374 (P^2), A113378 (P^3), A113384 (Q^2), A113387 (Q^3), A113389 (R), A113392 (R^2), A113394 (R^3).
%Y Cf. A114154 (R^3*Q^-2), A114155 (Q^-2*P^3).
%Y Cf. variants: A113340, A113350.
%K nonn,tabl
%O 0,2
%A _Paul D. Hanna_, Nov 14 2005
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