0,2

This triangle W is the column transform for triangles U=A136228 and V=A136230: W * [column k of U] = column k+1 of U, and W * [column k of V] = column k+1 of V, for k>=0.

Triangle W begins:

1;

3, 1;

15, 6, 1;

108, 48, 9, 1;

1036, 495, 99, 12, 1;

12569, 6338, 1323, 168, 15, 1;

185704, 97681, 21036, 2754, 255, 18, 1;

3247546, 1767845, 390012, 52204, 4950, 360, 21, 1;

65762269, 36839663, 8287041, 1128404, 108860, 8073, 483, 24, 1; ...

where column k of W = column 0 of W^(k+1) such that W = P^3

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^3 = column 0 of P^(3k+3) such that

column 0 of P^3 = column 0 of P shift up one row.

Also, this triangle W equals the matrix product:

W = V * [V shift down one row]

where triangle V = A136230 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; ...

and V shift down one row begins:

1;

1, 1;

2, 1, 1;

8, 5, 1, 1;

49, 35, 8, 1, 1;

414, 325, 80, 11, 1, 1;

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

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

Cf. A136221 (column 0); related tables: A136220 (P), A136225 (P^2), A136228 (U), A136230 (V), A136235 (W^2), A136238 (W^3); A136217, A136218.

Adjacent sequences: A136228 A136229 A136230 this_sequence A136232 A136233 A136234

Sequence in context: A048966 A104990 A089463 this_sequence A113389 A038553 A135896

nonn,tabl

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