A136231 Triangle W, read by rows, where column k of W = column 0 of W^(k+1) for k>=0 such that W equals the matrix cube of P = A136220 with column 0 of W = column 0 of P shift up one row.

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, 1515642725

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..45.

Example

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; ...

Prog

(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]}

Crossrefs

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

Sequence in context: A297704 A104990 A089463 * A113389 A038553 A282629

Adjacent sequences: A136228 A136229 A136230 * A136232 A136233 A136234

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 28 2008

Status

approved