A114151 Triangle, read by rows, given by the product R^-2*Q^3 = Q^-1*P^2 using triangular matrices P=A113370, Q=A113381, R=A113389.

1, 0, 1, 0, 3, 1, 0, 15, 6, 1, 0, 136, 66, 9, 1, 0, 1998, 1091, 153, 12, 1, 0, 41973, 24891, 3621, 276, 15, 1, 0, 1166263, 737061, 110637, 8482, 435, 18, 1, 0, 40747561, 27110418, 4176549, 323874, 16430, 630, 21, 1

Offset: 0

Author

Paul D. Hanna, Nov 15 2005

Keywords

• nonn
• tabl

Comments

Complementary to A114150, which gives R^2*Q^-1 = Q^3*P^-2.

Examples

			Triangle R^-2*Q^3 = Q^-1*P^2 begins:
1;
0,1;
0,3,1;
0,15,6,1;
0,136,66,9,1;
0,1998,1091,153,12,1;
0,41973,24891,3621,276,15,1; ...
Compare to R (A113389):
1;
3,1;
15,6,1;
136,66,9,1;
1998,1091,153,12,1;
41973,24891,3621,276,15,1; ...
Thus R^-2*Q^3 = Q^-1*P^2 equals R shift right one column.
		

Crossrefs

Cf. A113370 (P), A113381 (Q), A113389 (R); A114150 (R^2*Q^-1=Q^3*P^-2), A114152 (R^3*P^-1), A114153 (R^-1*P^3), A114154 (R^3*Q^-2), A114155 (Q^-2*P^3); A114156 (P^-1), A114158 (Q^-1), A114159 (R^-1).

Programs

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])); (Q^-1*P^2)[n+1,k+1]