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%I #5 Jun 13 2017 23:39:46
%S 1,5,1,45,8,1,635,120,11,1,12815,2556,231,14,1,343815,71548,6556,378,
%T 17,1,11651427,2508528,233706,13391,561,20,1,480718723,106427700,
%U 10069521,579047,23817,780,23,1
%N Triangle, read by rows, given by the product R^3*Q^-2 using triangular matrices Q=A113381, R=A113389.
%C Complementary to A114155, which gives Q^-2*P^3.
%e Triangle R^3*Q^-2 begins:
%e 1;
%e 5,1;
%e 45,8,1;
%e 635,120,11,1;
%e 12815,2556,231,14,1;
%e 343815,71548,6556,378,17,1; ...
%e Compare to Q (A113381):
%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 Thus R^3*Q^-2 equals Q shift left one column.
%o (PARI) T(n,k)=local(P,Q,R,W);P=Mat(1);for(m=2,n+1,W=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,W[i,j]=1,if(j==1, W[i,1]=1,W[i,j]=(P^(3*j-2))[i-j+1,1]));));P=W); Q=matrix(#P,#P,r,c,if(r>=c,(P^(3*c-1))[r-c+1,1])); R=matrix(#P,#P,r,c,if(r>=c,(P^(3*c))[r-c+1,1])); (R^3*Q^-2)[n+1,k+1]
%Y Cf. A113394 (R^3), A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114151 (R^-2*Q^3=Q^-1*P^2), A114152 (R^3*P^-1), A114153 (R^-1*P^3), A114155 (Q^-2*P^3); A114156 (P^-1), A114158 (Q^-1), A114159 (R^-1).
%K nonn,tabl
%O 0,2
%A _Paul D. Hanna_, Nov 15 2005
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