0,2

Triangle W=P^3=A136231 transforms column k of V into column k+1 of V. This triangle equals the matrix products: V = P^2 * [P shift right one column] and V = U * [U shift down one row] (see examples).

This triangle V begins:

1;

2, 1;

8, 5, 1;

49, 35, 8, 1;

414, 325, 80, 11, 1;

4529, 3820, 988, 143, 14, 1;

61369, 54800, 14696, 2200, 224, 17, 1;

996815, 932761, 257264, 39468, 4123, 323, 20, 1;

18931547, 18426632, 5198680, 812801, 86506, 6919, 440, 23, 1; ...

where column k of V = column 0 of P^(3k+2) and

triangle P = A136220 begins:

1;

1, 1;

3, 2, 1;

15, 10, 3, 1;

108, 75, 21, 4, 1;

1036, 753, 208, 36, 5, 1;

12569, 9534, 2637, 442, 55, 6, 1; ...

where column k of P^2 = column 0 of V^(k+1).

Also, this triangle V equals the matrix product:

V = P^2 * [P shift right one column]

where P^2 = A136225 begins:

1;

2, 1;

8, 4, 1;

49, 26, 6, 1;

414, 232, 54, 8, 1;

4529, 2657, 629, 92, 10, 1;

61369, 37405, 9003, 1320, 140, 12, 1; ...

and P shift right one column begins:

1;

0, 1;

0, 1, 1;

0, 3, 2, 1;

0, 15, 10, 3, 1;

0, 108, 75, 21, 4, 1;

0, 1036, 753, 208, 36, 5, 1; ...

Also, this triangle V equals the matrix product:

V = U * [U shift down one row]

where triangle U = A136228 begins:

1;

1, 1;

3, 4, 1;

15, 24, 7, 1;

108, 198, 63, 10, 1;

1036, 2116, 714, 120, 13, 1; ...

and U shift down one row begins:

1;

1, 1;

1, 1, 1;

3, 4, 1, 1;

15, 24, 7, 1, 1;

108, 198, 63, 10, 1, 1;

1036, 2116, 714, 120, 13, 1, 1; ...

(PARI) {T(n, k)=local(P=Mat(1), U=Mat(1), V=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])))); U=P*PShR^2; V=P^2*PShR; U=matrix(#P+1, #P+1, r, c, if(r>=c, if(r<#P+1, U[r, c], if(c==1, (P^3)[ #P, 1], (P^(3*c-1))[r-c+1, 1])))); V=matrix(#P+1, #P+1, r, c, if(r>=c, if(r<#P+1, V[r, c], if(c==1, (P^3)[ #P, 1], (P^(3*c-2))[r-c+1, 1])))); P=matrix(#U, #U, r, c, if(r>=c, if(r<#R, P[r, c], (U^c)[r-c+1, 1]))))); V[n+1, k+1]}

Cf. A136226 (column 0), A136229 (column 1); related tables: A136220 (P), A136225 (P^2), A136230 (V), A136231 (W=P^3), A136234 (V^2), A136237 (V^3); A136217, A136218.

Adjacent sequences: A136227 A136228 A136229 this_sequence A136231 A136232 A136233

Sequence in context: A136225 A089460 A135520 this_sequence A004732 A011244 A008517

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 28 2008