0,4

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.

Column k of R = column 2 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.

Triangle R 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;

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

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

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

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

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

and 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; ...

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

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

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

(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]}

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).

Sequence in context: A130534 A107416 A105613 this_sequence A130850 A075263 A130405

Adjacent sequences: A135891 A135892 A135893 this_sequence A135895 A135896 A135897

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 15 2007