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%I #17 Aug 13 2019 10:54:41
%S 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,
%T 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,
%U 115975,3425,92,1,1,0,0,0,0,0,1,678570,17722,491,36,1,1,0,0,0,0,0,1
%N Generalized Bell numbers; square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 1.
%H E. A. Enneking and J. C. Ahuja, <a href="http://www.fq.math.ca/Scanned/14-1/enneking.pdf">Generalized Bell numbers</a>, Fib. Quart., 14 (1976), 67-73.
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/SetPartitions">Set partitions</a>
%F E.g.f.: exp(exp(x)*(1-Gamma(k,x)/Gamma(k))); Gamma(k,x) the incomplete Gamma function.
%e Array starts:
%e [k= 1 2 3 4 5]
%e [n=0] 1, 1, 1, 1, 1,
%e [n=1] 1, 0, 0, 0, 0,
%e [n=2] 2, 1, 0, 0, 0,
%e [n=3] 5, 1, 1, 0, 0,
%e [n=4] 15, 4, 1, 1, 0,
%e [n=5] 52, 11, 1, 1, 1,
%e [n=6] 203, 41, 11, 1, 1,
%e [n=7] 877, 162, 36, 1, 1,
%e [n=8] 4140, 715, 92, 36, 1,
%e A000110,A000296,A006505,A057837,A057814, ...
%p egf := k -> exp(exp(x)*(1-GAMMA(k,x)/GAMMA(k)));
%p T := (n,k) -> n!*coeff(series(egf(k),x,n+1),x,n):
%p seq(print(seq(T(n,k),k=1..8)),n=0..8);
%t egf[k_] := Exp[Exp[x] (1 - Gamma[k, x]/Gamma[k])];
%t T[n_, k_] := n! SeriesCoefficient[egf[k], {x, 0, n}];
%t Table[T[n-k+1, k], {n, 0, 11}, {k, 1, n+1}] // Flatten (* _Jean-François Alcover_, Aug 13 2019 *)
%Y Cf. A000110, A000296, A006505, A057837, A057814, A097147.
%Y Row sums are A097147 for n >= 1.
%K nonn,tabl
%O 1,4
%A _Peter Luschny_, Apr 05 2011
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