A109282 Triangle T, read by rows, that satisfies: T(n,k) = [T^3](n-1,k) for n>k+1>=1, with T(n,n) = 1 and T(n+1,n) = n+1 for n>=0, where T^3 is the matrix cube of T.

1, 1, 1, 3, 2, 1, 15, 6, 3, 1, 96, 36, 9, 4, 1, 735, 258, 63, 12, 5, 1, 6447, 2190, 492, 96, 15, 6, 1, 63120, 20988, 4545, 804, 135, 18, 7, 1, 677739, 222042, 46935, 7980, 1200, 180, 21, 8, 1, 7878921, 2554890, 530562, 87960, 12675, 1686, 231, 24, 9, 1

Offset

0,4

Links

Table of n, a(n) for n=0..54.

Formula

T^(m+3) = SHIFT_UP(T^(m+1) - T^m) - D*T^m for all m where diagonal matrix D = [0, 1, 2, 3, ...] and SHIFT_UP shifts each column up 1 row.

Example

Triangle T begins:
1;
1,1;
3,2,1;
15,6,3,1;
96,36,9,4,1;
735,258,63,12,5,1;
6447,2190,492,96,15,6,1;
63120,20988,4545,804,135,18,7,1;
677739,222042,46935,7980,1200,180,21,8,1; ...
Matrix cube T^3 starts:
1;
3,1;
15,6,1;
96,36,9,1;
735,258,63,12,1;
6447,2190,492,96,15,1; ...
which equals SHIFT_UP(T) - D where
D is the diagonal matrix [0,1,2,3,...].

Prog

(PARI) {T(n, k)=local(M=matrix(n+3, n+3)); M=M^0; for(i=1, n, M=matrix(n+3, n+3, r, c, if(r>=c, if(r==c, 1, if(r==c+1, c, (M^3)[r-1, c]))))); return((M^3)[n+1, k+1])}

Crossrefs

Cf. A109152, A109283 (column 0), A109284 (column 1), A109285 (column 2), A109286 (row sums).

Sequence in context: A342217 A111548 A140709 * A135902 A135876 A136217

Adjacent sequences: A109279 A109280 A109281 * A109283 A109284 A109285

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 24 2005

Status

approved