A135900 - OEIS

%I #2 Mar 30 2012 18:37:08

%S 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,

%T 970,3472,1945,525,90,11,1,1,8390,34380,20340,5733,1026,132,13,1,1,

%U 86796,399463,247066,72030,13305,1771,182,15,1,1,1049546,5344770,3430936

%N 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.

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 2, 3, 1, 1;

%e 6, 12, 5, 1, 1;

%e 25, 63, 30, 7, 1, 1;

%e 138, 421, 220, 56, 9, 1, 1;

%e 970, 3472, 1945, 525, 90, 11, 1, 1;

%e 8390, 34380, 20340, 5733, 1026, 132, 13, 1, 1;

%e 86796, 399463, 247066, 72030, 13305, 1771, 182, 15, 1, 1; ...

%e This triangle equals matrix product R^-1*Q,

%e which equals triangle R shifted down one row,

%e where P = A135880 begins:

%e 1;

%e 1, 1;

%e 2, 2, 1;

%e 6, 7, 3, 1;

%e 25, 34, 15, 4, 1;

%e 138, 215, 99, 26, 5, 1;

%e 970, 1698, 814, 216, 40, 6, 1; ...

%e and Q = P^2 = A135885 begins:

%e 1;

%e 2, 1;

%e 6, 4, 1;

%e 25, 20, 6, 1;

%e 138, 126, 42, 8, 1;

%e 970, 980, 351, 72, 10, 1;

%e 8390, 9186, 3470, 748, 110, 12, 1; ...

%e and R = A135894 begins:

%e 1;

%e 1, 1;

%e 2, 3, 1;

%e 6, 12, 5, 1;

%e 25, 63, 30, 7, 1;

%e 138, 421, 220, 56, 9, 1;

%e 970, 3472, 1945, 525, 90, 11, 1; ...

%e where column k of R equals column 0 of P^(2k+1),

%e and column k of Q=P^2 equals column 0 of P^(2k+2), for k>=0.

%o (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]}

%Y Cf. A135880 (P), A135885 (Q=P^2), A135894 (R); A135898 (P^-1*R), A135899 (P*R^-1*P).

%K nonn,tabl

%O 0,7

%A _Paul D. Hanna_, Dec 15 2007