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

%I A182931 #19 Jan 05 2025 19:51:39
%S A182931 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 A182931 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 A182931 115975,3425,92,1,1,0,0,0,0,0,1,678570,17722,491,36,1,1,0,0,0,0,0,1
%N A182931 Generalized Bell numbers; square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 1.
%H A182931 E. A. Enneking and J. C. Ahuja, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-1/enneking.pdf">Generalized Bell numbers</a>, Fib. Quart., 14 (1976), 67-73.
%H A182931 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/SetPartitions">Set partitions</a>
%F A182931 E.g.f.: exp(exp(x)*(1-Gamma(k,x)/Gamma(k))); Gamma(k,x) the incomplete Gamma function.
%e A182931 Array starts:
%e A182931 [k=      1       2       3       4       5]
%e A182931 [n=0]    1,      1,      1,      1,      1,
%e A182931 [n=1]    1,      0,      0,      0,      0,
%e A182931 [n=2]    2,      1,      0,      0,      0,
%e A182931 [n=3]    5,      1,      1,      0,      0,
%e A182931 [n=4]   15,      4,      1,      1,      0,
%e A182931 [n=5]   52,     11,      1,      1,      1,
%e A182931 [n=6]  203,     41,     11,      1,      1,
%e A182931 [n=7]  877,    162,     36,      1,      1,
%e A182931 [n=8] 4140,    715,     92,     36,      1,
%e A182931    A000110,A000296,A006505,A057837,A057814, ...
%p A182931 egf := k -> exp(exp(x)*(1-GAMMA(k,x)/GAMMA(k)));
%p A182931 T := (n,k) -> n!*coeff(series(egf(k),x,n+1),x,n):
%p A182931 seq(print(seq(T(n,k),k=1..8)),n=0..8);
%t A182931 egf[k_] := Exp[Exp[x] (1 - Gamma[k, x]/Gamma[k])];
%t A182931 T[n_, k_] := n! SeriesCoefficient[egf[k], {x, 0, n}];
%t A182931 Table[T[n-k+1, k], {n, 0, 11}, {k, 1, n+1}] // Flatten (* _Jean-François Alcover_, Aug 13 2019 *)
%Y A182931 Cf. A000110, A000296, A006505, A057837, A057814, A097147.
%Y A182931 Row sums are A097147 for n >= 1.
%K A182931 nonn,tabl
%O A182931 1,4
%A A182931 _Peter Luschny_, Apr 05 2011