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A135896 Triangle, read by rows, equal to R^3, the matrix cube of R = A135894.

%I A135896 #2 Mar 30 2012 18:37:08
%S A135896 1,3,1,15,9,1,99,81,15,1,814,816,195,21,1,8057,9366,2625,357,27,1,
%T A135896 93627,122148,38270,6006,567,33,1,1252752,1795481,611525,105910,11439,
%U A135896 825,39,1,19003467,29478724,10721093,1996988,236430,19404,1131,45,1,322722064
%N A135896 Triangle, read by rows, equal to R^3, the matrix cube of R = A135894.
%C A135896 Triangle P = A135880 is defined by: column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift left.
%F A135896 Column k of R^3 = column 2 of P^(2k+1) for k>=0 where triangle P = A135880; column 0 of R^3 = column 2 of P; column 1 of R^3 = column 2 of P^3; column 2 of R^3 = column 2 of P^5.
%e A135896 Triangle R^3 begins:
%e A135896 1;
%e A135896 3, 1;
%e A135896 15, 9, 1;
%e A135896 99, 81, 15, 1;
%e A135896 814, 816, 195, 21, 1;
%e A135896 8057, 9366, 2625, 357, 27, 1;
%e A135896 93627, 122148, 38270, 6006, 567, 33, 1;
%e A135896 1252752, 1795481, 611525, 105910, 11439, 825, 39, 1;
%e A135896 19003467, 29478724, 10721093, 1996988, 236430, 19404, 1131, 45, 1; ...
%e A135896 where R = A135894 begins:
%e A135896 1;
%e A135896 1, 1;
%e A135896 2, 3, 1;
%e A135896 6, 12, 5, 1;
%e A135896 25, 63, 30, 7, 1;
%e A135896 138, 421, 220, 56, 9, 1;
%e A135896 970, 3472, 1945, 525, 90, 11, 1; ...
%e A135896 where column k of R = column 0 of P^(2k+1)
%e A135896 and P = A135880 begins:
%e A135896 1;
%e A135896 1, 1;
%e A135896 2, 2, 1;
%e A135896 6, 7, 3, 1;
%e A135896 25, 34, 15, 4, 1;
%e A135896 138, 215, 99, 26, 5, 1;
%e A135896 970, 1698, 814, 216, 40, 6, 1; ...
%e A135896 where column k of P equals column 0 of R^(k+1).
%o A135896 (PARI) {T(n,k)=local(P=Mat(1),R=Mat(1),PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));(R^3)[n+1,k+1]}
%Y A135896 Cf. A135883 (column 0); A135894 (R), A135880 (P), A135888 (P^3), A135892 (P^5).
%K A135896 nonn,tabl
%O A135896 0,2
%A A135896 _Paul D. Hanna_, Dec 15 2007