A135898 Triangle, read by rows equal to the matrix product P^-1*R, where P = A135880 and R = A135894; P^-1*R equals triangle P shifted right one column.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 7, 3, 1, 0, 25, 34, 15, 4, 1, 0, 138, 215, 99, 26, 5, 1, 0, 970, 1698, 814, 216, 40, 6, 1, 0, 8390, 16220, 8057, 2171, 400, 57, 7, 1, 0, 86796, 182714, 93627, 25628, 4740, 666, 77, 8, 1, 0, 1049546, 2378780, 1252752, 348050, 64805

Offset: 0

Author

Paul D. Hanna, Dec 15 2007

Keywords

• nonn
• tabl

Examples

			Triangle begins:
1;
0, 1;
0, 1, 1;
0, 2, 2, 1;
0, 6, 7, 3, 1;
0, 25, 34, 15, 4, 1;
0, 138, 215, 99, 26, 5, 1;
0, 970, 1698, 814, 216, 40, 6, 1;
0, 8390, 16220, 8057, 2171, 400, 57, 7, 1;
0, 86796, 182714, 93627, 25628, 4740, 666, 77, 8, 1; ...
This triangle equals matrix product P^-1*R,
which equals triangle P shifted right one column,
where P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
and Q = P^2 = A135885 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1; ...
and R = A135894 begins:
1;
1, 1;
2, 3, 1;
6, 12, 5, 1;
25, 63, 30, 7, 1;
138, 421, 220, 56, 9, 1;
970, 3472, 1945, 525, 90, 11, 1; ...
where column k of R equals column 0 of P^(2k+1),
and column k of Q=P^2 equals column 0 of P^(2k+2), for k>=0.
		

Crossrefs

Cf. A135880 (P), A135885 (Q=P^2), A135894 (R); A135899 (P*R^-1*P), A135900 (R^-1*Q).

Programs

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])))));(P^-1*R)[n+1,k+1]}