A182931 Generalized Bell numbers; square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 1.

1, 1, 1, 2, 0, 1, 5, 1, 0, 1, 15, 1, 0, 0, 1, 52, 4, 1, 0, 0, 1, 203, 11, 1, 0, 0, 0, 1, 877, 41, 1, 1, 0, 0, 0, 1, 4140, 162, 11, 1, 0, 0, 0, 0, 1, 21147, 715, 36, 1, 1, 0, 0, 0, 0, 1, 115975, 3425, 92, 1, 1, 0, 0, 0, 0, 0, 1, 678570, 17722, 491, 36, 1, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Peter Luschny, Apr 05 2011

Keywords

• nonn
• tabl

Examples

			Array starts:
[k=      1       2       3       4       5]
[n=0]    1,      1,      1,      1,      1,
[n=1]    1,      0,      0,      0,      0,
[n=2]    2,      1,      0,      0,      0,
[n=3]    5,      1,      1,      0,      0,
[n=4]   15,      4,      1,      1,      0,
[n=5]   52,     11,      1,      1,      1,
[n=6]  203,     41,     11,      1,      1,
[n=7]  877,    162,     36,      1,      1,
[n=8] 4140,    715,     92,     36,      1,
   A000110,A000296,A006505,A057837,A057814, ...
		

Links

• E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
• Peter Luschny, Set partitions

Crossrefs

Cf. A000110, A000296, A006505, A057837, A057814, A097147.

Row sums are A097147 for n >= 1.

Programs

Maple

egf := k -> exp(exp(x)*(1-GAMMA(k,x)/GAMMA(k)));
T := (n,k) -> n!*coeff(series(egf(k),x,n+1),x,n):
seq(print(seq(T(n,k),k=1..8)),n=0..8);

Mathematica

egf[k_] := Exp[Exp[x] (1 - Gamma[k, x]/Gamma[k])];
T[n_, k_] := n! SeriesCoefficient[egf[k], {x, 0, n}];
Table[T[n-k+1, k], {n, 0, 11}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 13 2019 *)

Formula

E.g.f.: exp(exp(x)*(1-Gamma(k,x)/Gamma(k))); Gamma(k,x) the incomplete Gamma function.