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A114152 Triangle, read by rows, given by the product R^3*P^-1 using triangular matrices P=A113370, R=A113389.

%I A114152 #5 Jun 13 2017 23:39:48
%S A114152 1,8,1,84,14,1,1296,252,20,1,27850,5957,510,26,1,784146,179270,16180,
%T A114152 858,32,1,27630378,6641502,623115,34125,1296,38,1,1177691946,
%U A114152 294524076,28470525,1599091,61952,1824,44,1
%N A114152 Triangle, read by rows, given by the product R^3*P^-1 using triangular matrices P=A113370, R=A113389.
%C A114152 Complementary to A114153, which gives R^-1*P^3.
%e A114152 Triangular matrix R^3*P^-1 begins:
%e A114152 1;
%e A114152 8,1;
%e A114152 84,14,1;
%e A114152 1296,252,20,1;
%e A114152 27850,5957,510,26,1;
%e A114152 784146,179270,16180,858,32,1;
%e A114152 27630378,6641502,623115,34125,1296,38,1; ...
%e A114152 Compare to P^2 (A113374):
%e A114152 1;
%e A114152 2,1;
%e A114152 6,8,1;
%e A114152 37,84,14,1;
%e A114152 429,1296,252,20,1;
%e A114152 7629,27850,5957,510,26,1; ...
%e A114152 Thus R^3*P^-1 equals P^2 shift left one column.
%o A114152 (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*P^-1)[n+1,k+1]
%Y A114152 Cf. A113374 (P^2), A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114151 (R^-2*Q^3=Q^-1*P^2), A114153 (R^-1*P^3), A114154 (R^3*Q^-2), A114155 (Q^-2*P^3); A114156 (P^-1), A114158 (Q^-1), A114159 (R^-1).
%K A114152 nonn,tabl
%O A114152 0,2
%A A114152 _Paul D. Hanna_, Nov 15 2005