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%I #7 Jun 13 2017 23:30:47
%S 1,3,1,15,6,1,136,66,9,1,1998,1091,153,12,1,41973,24891,3621,276,15,1,
%T 1166263,737061,110637,8482,435,18,1,40747561,27110418,4176549,323874,
%U 16430,630,21,1,1726907675,1199197442,188802141,14813844,751920,28221
%N Triangle R, read by rows, such that R^3 transforms column k of R^3 into column k+1 of R^3, so that column k of R^3 equals column 0 of R^(3*k+3), where R^3 denotes the matrix cube of R.
%C Related matrix products: identity R^-2*Q^3 = Q^-1*P^2 (A114151) and R^-1*P^3 (A114153).
%F Let [R^m]_k denote column k of matrix power R^m,
%F so that triangular matrix R may be defined by
%F [R]_k = [P^(3*k+3)]_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 Q = A113381 by
%F [Q]_k = [P^(3*k+2)]_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 R begins:
%e 1;
%e 3,1;
%e 15,6,1;
%e 136,66,9,1;
%e 1998,1091,153,12,1;
%e 41973,24891,3621,276,15,1;
%e 1166263,737061,110637,8482,435,18,1;
%e 40747561,27110418,4176549,323874,16430,630,21,1;
%e 1726907675,1199197442,188802141,14813844,751920,28221,861,24,1;
%e Matrix cube R^3 (A113394) starts:
%e 1;
%e 9,1;
%e 99,18,1;
%e 1569,360,27,1;
%e 34344,9051,783,36,1;
%e 980487,284148,26820,1368,45,1; ...
%e where R^3 transforms column k of R^3 into column k+1:
%e at k=0, [R^3]*[1,9,99,1569,...] = [1,18,360,9051,...];
%e at k=1, [R^3]*[1,18,360,9051,..] = [1,27,783,26820,..].
%o (PARI) R(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+3))[n-k+1,1]
%Y Cf. A113379 (column 0), A113390 (column 1), A113391 (column 2).
%Y Cf. A113370 (P), A113374 (P^2), A113378 (P^3), A113381 (Q), A113384 (Q^2), A113387 (Q^3), A113392 (R^2), A113394 (R^3).
%Y Cf. A114151 (R^-2*Q^3 = Q^-1*P^2), A114153 (R^-1*P^3).
%Y Cf. variants: A113340, A113350.
%K nonn,tabl
%O 0,2
%A _Paul D. Hanna_, Nov 14 2005
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