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
KEYWORD
sign,tabl
AUTHOR
N. J. A. Sloane, Jan 21 2009
EXTENSIONS
Corrected and extended by Alois P. Heinz, Oct 23 2009
STATUS
approved