%I #4 Mar 30 2012 18:37:08

%S 1,1,1,2,3,1,6,12,5,1,25,63,30,7,1,138,421,220,56,9,1,970,3472,1945,

%T 525,90,11,1,8390,34380,20340,5733,1026,132,13,1,86796,399463,247066,

%U 72030,13305,1771,182,15,1,1049546,5344770,3430936,1028076,194646,26565,2808

%N Triangle R, read by rows, where column k of R equals column 0 of P^(2k+1) where P=A135880.

%C Triangle P = A135880 is defined by: column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift left.

%F Column k of R = column 0 of P^(2k+1) for k>=0 where triangle P = A135880; column 0 of R = column 0 of P; column 1 of R = column 0 of P^3; column 2 of R = column 0 of P^5. See more formulas relating triangles P, Q and R, in entry A135880.

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

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

%e 1049546, 5344770, 3430936, 1028076, 194646, 26565, 2808, 240, 17, 1;

%e 14563135, 81097517, 53741404, 16477041, 3182778, 442948, 47801, 4185, 306, 19, 1; ...

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

%e and 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 where column k of P equals column 0 of R^(k+1).

%e The matrix product P^-1*R = A135898 = P (shifted right one column);

%e the matrix product R^-1*P^2 = A135900 = R (shifted down one row).

%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[n+1,k+1]}

%Y Cf. A135881 (column 0), A135889 (column 1); A135880 (P), A135885 (Q=P^2), A135895 (R^2), A135896 (R^3), A135897 (R^4); A135888 (P^3) A135892 (P^5); A135898 (P^-1*R), A135899 (P*R^-1*P), A135900 (R^-1*Q).

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Dec 15 2007

%E Typo in formula corrected by _Paul D. Hanna_, Mar 26 2010