A135900 Triangle, read by rows equal to the matrix product R^-1*Q, where Q = A135885 and R = A135894; R^-1*Q equals triangle R shifted down one row.

1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 6, 12, 5, 1, 1, 25, 63, 30, 7, 1, 1, 138, 421, 220, 56, 9, 1, 1, 970, 3472, 1945, 525, 90, 11, 1, 1, 8390, 34380, 20340, 5733, 1026, 132, 13, 1, 1, 86796, 399463, 247066, 72030, 13305, 1771, 182, 15, 1, 1, 1049546, 5344770, 3430936

Offset: 0

Author

Paul D. Hanna, Dec 15 2007

Keywords

• nonn
• tabl

Examples

			Triangle begins:
1;
1, 1;
1, 1, 1;
2, 3, 1, 1;
6, 12, 5, 1, 1;
25, 63, 30, 7, 1, 1;
138, 421, 220, 56, 9, 1, 1;
970, 3472, 1945, 525, 90, 11, 1, 1;
8390, 34380, 20340, 5733, 1026, 132, 13, 1, 1;
86796, 399463, 247066, 72030, 13305, 1771, 182, 15, 1, 1; ...
This triangle equals matrix product R^-1*Q,
which equals triangle R shifted down one row,
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); A135898 (P^-1*R), A135899 (P*R^-1*P).

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