login
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.
11
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.
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jan 28 2008
STATUS
approved