A135899 Triangle, read by rows equal to the matrix product P*R^-1*P, where P = A135880 and R = A135894; P*R^-1*P equals triangle Q=A135885 shifted down one row.

1, 1, 1, 2, 1, 1, 6, 4, 1, 1, 25, 20, 6, 1, 1, 138, 126, 42, 8, 1, 1, 970, 980, 351, 72, 10, 1, 1, 8390, 9186, 3470, 748, 110, 12, 1, 1, 86796, 101492, 39968, 8936, 1365, 156, 14, 1, 1, 1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1, 1, 14563135, 18868652

Offset

0,4

Links

Table of n, a(n) for n=0..56.

Example

Triangle begins:
1;
1, 1;
2, 1, 1;
6, 4, 1, 1;
25, 20, 6, 1, 1;
138, 126, 42, 8, 1, 1;
970, 980, 351, 72, 10, 1, 1;
8390, 9186, 3470, 748, 110, 12, 1, 1;
86796, 101492, 39968, 8936, 1365, 156, 14, 1, 1;
1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1, 1; ...
This triangle equals matrix product P*R^-1*P,
which equals triangle Q 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.

Prog

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

Crossrefs

Cf. A135880 (P), A135885 (Q=P^2), A135894 (R); A135898 (P^-1*R), A135900 (R^-1*Q).

Sequence in context: A257895 A186023 A103880 * A327816 A047920 A350269

Adjacent sequences: A135896 A135897 A135898 * A135900 A135901 A135902

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 15 2007

Status

approved