A144633 Triangle of 3-restricted Stirling numbers of the first kind (T(n,k), 0 <= k <= n), read by rows.

1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -5, 11, -6, 1, 0, 10, -45, 35, -10, 1, 0, 35, 175, -210, 85, -15, 1, 0, -910, -315, 1225, -700, 175, -21, 1, 0, 11935, -6265, -5670, 5565, -1890, 322, -28, 1, 0, -134750, 139755, -5005, -39270, 19425, -4410, 546, -36, 1

Offset

0,8

Comments

Definition: take the triangle in A144385, write it as an (infinite) upper triangular square matrix, invert it and transpose it.

The Bell transform of A144636(n+1). Also the inverse Bell transform of the sequence "g(n) = 1 if n<3 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

References

J. Y. Choi and J. D. H. Smith, On the combinatorics of multi-restricted numbers, Ars. Com., 75(2005), pp. 44-63.

Links

Alois P. Heinz, Rows n = 0..140, flattened

J. Y. Choi and J. D. H. Smith, The Tri-restricted Numbers and Powers of Permutation Representations, J. Comb. Math. Comb. Comp. 42 (2002), 113-125.

J. Y. Choi and J. D. H. Smith, On the Unimodality and Combinatorics of the Bessel Numbers, Discrete Math., 264 (2003), 45-53.

J. Y. Choi et al., Reciprocity for multirestricted Stirling numbers, J. Combin. Theory 113 A (2006), 1050-1060.

Example

Triangle begins:
1;
0,    1;
0,   -1,    1;
0,    2,   -3,    1;
0,   -5,   11,   -6,    1;
0,   10,  -45,   35,  -10,   1;
0,   35,  175, -210,   85, -15,   1;
0, -910, -315, 1225, -700, 175, -21,  1;

Maple

A:= proc(n, k) option remember; if n=k then 1 elif k<n or n<1 then 0 else A(n-1, k-1) +(k-1) *A(n-1, k-2) +(k-1) *(k-2) *A(n-1, k-3)/2 fi end:
M:= proc(n) option remember; Matrix(n+1, (i, j)-> A(i-1, j-1))^(-1) end:
T:= (n, k)-> M(n+1)[k+1, n+1]:
seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Oct 23 2009

Mathematica

max = 10; t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; (0 <= k <= 3*n) := t[n, k] = t[n-1, k-1] + (k-1)*t[n-1, k-2] + (1/2)*(k-1)*(k-2)*t[n-1, k-3]; t[_, _] = 0; A144633 = Table[t[n, k], {n, 0, max}, {k, 0, max}] // Inverse // Transpose; Table[A144633[[n, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

Prog

(Sage) # uses[bell_matrix from A264428]
bell_matrix(lambda n: A144636(n+1), 10) # Peter Luschny, Jan 18 2016

Crossrefs

For another version of this triangle see A144634.

Columns give A144636-A144639.

Cf. A144402.

Sequence in context: A298753 A173050 A172380 * A352366 A264428 A256550

Adjacent sequences: A144630 A144631 A144632 * A144634 A144635 A144636

Keyword

sign,tabl

Author

N. J. A. Sloane, Jan 21 2009

Extensions

Corrected and extended by Alois P. Heinz, Oct 23 2009

Status

approved