A135885 Triangle Q, read by rows, where column k of Q equals column 0 of Q^(k+1) and Q is equal to the matrix square of integer triangle P = A135880 such that column 0 of Q equals column 0 of P shift left.

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

Offset: 0

Author

Paul D. Hanna, Dec 15 2007

Keywords

• nonn
• tabl

Examples

			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).
		

Crossrefs

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).

Programs

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]}

Formula

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