0,2

See formulas relating triangles P, Q and R, in entry A135880.

Triangle Q = P^2 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;

86796, 101492, 39968, 8936, 1365, 156, 14, 1;

1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1;

14563135, 18868652, 7906598, 1861416, 298830, 36028, 3451, 272, 18, 1;

228448504, 308478492, 132426050, 31785380, 5193982, 637390, 62230, 5016, 342, 20, 1; ...

where column k of Q equals column 0 of Q^(k+1) for k>=0.

Related triangle 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 Q equals column 0 of P^(2k+2)

such that column 0 of P^2 equals column 0 of P shift left.

The matrix product P*R^-1*P = A135899 = Q (shifted down one row),

where 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; ...

in which column k of R equals column 0 of P^(2k+1).

(PARI) {T(n, k)=local(P=Mat(1), R, 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^2)[n+1, k+1]}

Cf. columns: A135881, A135886, A135887; related tables: A135880 (P), A135894 (R), A135891 (Q^2), A135893 (Q^3); A135898 (P^-1*R), A135899 (P*R^-1*P), A135900 (R^-1*Q).

Sequence in context: A167560 A132159 A112356 this_sequence A162312 A141715 A098697

Adjacent sequences: A135882 A135883 A135884 this_sequence A135886 A135887 A135888

nonn,tabl

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