A135902 Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^3 shift right.

1, 1, 1, 3, 2, 1, 15, 8, 3, 1, 102, 47, 15, 4, 1, 860, 356, 102, 24, 5, 1, 8548, 3252, 860, 186, 35, 6, 1, 97094, 34448, 8548, 1736, 305, 48, 7, 1, 1234324, 412546, 97094, 18754, 3130, 465, 63, 8, 1, 17302880, 5488222, 1234324, 228658, 36630, 5212, 672, 80, 9, 1

Offset

0,4

Comments

This is a variant of permutation triangle P = A094587, A094587(n,k) = n!/k!, which may be defined by: triangular matrix P where column k of P = column 0 of P^(k+1), with column 0 of P = column 0 of P^2 shift right.

Links

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

Formula

Column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.

Example

Triangle T begins:
1;
1, 1;
3, 2, 1;
15, 8, 3, 1;
102, 47, 15, 4, 1;
860, 356, 102, 24, 5, 1;
8548, 3252, 860, 186, 35, 6, 1;
97094, 34448, 8548, 1736, 305, 48, 7, 1;
1234324, 412546, 97094, 18754, 3130, 465, 63, 8, 1;
17302880, 5488222, 1234324, 228658, 36630, 5212, 672, 80, 9, 1; ...
where column k of T = column 0 of T^(k+1)
with column 0 of T = column 0 of T^3 shift right.
Matrix square of T, T^2, begins:
1;
2, 1;
8, 4, 1;
47, 22, 6, 1;
356, 156, 42, 8, 1;
3252, 1343, 351, 68, 10, 1;
34448, 13493, 3415, 656, 100, 12, 1; ...
where column 0 of T^2 = column 1 of T.
Matrix cube of T, T^3, begins:
1;
3, 1;
15, 6, 1;
102, 42, 9, 1;
860, 351, 81, 12, 1;
8548, 3415, 807, 132, 15, 1;
97094, 37795, 8967, 1530, 195, 18, 1; ...
where column 0 of T^3 = column 2 of T = column 0 of T shift left;
also, column 1 of T^3 = column 2 of T^2.

Prog

(PARI) T(n, k)=if(k>n || n<0 || k<0, 0, if(k==n, 1, if(k==0, T(n+1, 2), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ); )

Crossrefs

Cf. columns: A135903, A135904, A135905; variants: A091351, A094587.

Sequence in context: A111548 A140709 A109282 * A135876 A136217 A166884

Adjacent sequences: A135899 A135900 A135901 * A135903 A135904 A135905

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 18 2007

Status

approved