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%I #6 Jun 13 2017 23:21:12
%S 1,1,1,1,4,1,1,28,7,1,1,326,91,10,1,1,5702,1722,190,13,1,1,136724,
%T 43764,4945,325,16,1,1,4226334,1415799,163705,10751,496,19,1,1,
%U 161385532,56096733,6617605,437723,19896,703,22,1,1,7378504140,2644883675
%N Triangle P, read by rows, such that P^3 transforms column k of P into column k+1 of P, so that column k of P equals column 0 of P^(3*k+1), where P^3 denotes the matrix cube of P.
%C Triangle A114150 illustrates the identity: R^2*Q^-1 = Q^3*P^-2.
%C See also A114152 for the matrix product: R^3*P^-1.
%F Let [P^m]_k denote column k of matrix power P^m,
%F so that triangular matrix P may be defined by
%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 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 P begins:
%e 1;
%e 1,1;
%e 1,4,1;
%e 1,28,7,1;
%e 1,326,91,10,1;
%e 1,5702,1722,190,13,1;
%e 1,136724,43764,4945,325,16,1;
%e 1,4226334,1415799,163705,10751,496,19,1;
%e 1,161385532,56096733,6617605,437723,19896,703,22,1;
%e 1,7378504140,2644883675,317416204,21179483,960696,33136,946,25,1;
%e Matrix cube P^3 (A113378) starts:
%e 1;
%e 3,1;
%e 15,12,1;
%e 136,168,21,1;
%e 1998,3190,483,30,1;
%e 41973,80136,13615,960,39,1; ...
%e where P^3 transforms column k of P into column k+1 of P:
%e at k=0, [P^3]*[1,1,1,1,1,...] = [1,4,28,326,5702,...];
%e at k=1, [P^3]*[1,4,28,326,5702,...] = [1,7,91,1722,43764,...].
%o (PARI) P(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[n+1,k+1]
%Y Cf. A113371 (column 1), A113372 (column 2), A113373 (column 3).
%Y Cf. A113374 (P^2), A113378 (P^3), A113381 (Q), A113384 (Q^2), A113387 (Q^3), A113389 (R), A113392 (R^2), A113394 (R^3), A114156 (P^-1).
%Y Cf. A114150 (R^2*Q^-1=Q^3*P^-2), A114152 (R^3*P^-1).
%Y Cf. variants: A113340, A113350.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Nov 14 2005
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