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A368759 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * (1 + Sum_{j=0..n} j^k/j!).

%I A368759 #13 Jan 05 2024 07:57:18
%S A368759 2,1,3,1,2,7,1,2,6,22,1,2,8,21,89,1,2,12,33,88,446,1,2,20,63,148,445,
%T A368759 2677,1,2,36,141,316,765,2676,18740,1,2,68,351,820,1705,4626,18739,
%U A368759 149921,1,2,132,933,2428,4725,10446,32431,149920,1349290,1,2,260,2583,7828,15265,29646,73465,259512,1349289,13492901
%N A368759 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * (1 + Sum_{j=0..n} j^k/j!).
%H A368759 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%H A368759 Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>.
%F A368759 T(0,k) = 1 + 0^k and T(n,k) = n^k + n * T(n-1,k) for n>0.
%F A368759 T(n,k) = n! + A337085(n,k).
%F A368759 E.g.f. of column k: (1+ B_k(x) * exp(x)) / (1-x), where B_n(x) = Bell polynomials.
%e A368759 Square array begins:
%e A368759      2,    1,    1,     1,     1,     1,      1, ...
%e A368759      3,    2,    2,     2,     2,     2,      2, ...
%e A368759      7,    6,    8,    12,    20,    36,     68, ...
%e A368759     22,   21,   33,    63,   141,   351,    933, ...
%e A368759     89,   88,  148,   316,   820,  2428,   7828, ...
%e A368759    446,  445,  765,  1705,  4725, 15265,  54765, ...
%e A368759   2677, 2676, 4626, 10446, 29646, 99366, 375246, ...
%o A368759 (PARI) T(n, k) = n!*(1+sum(j=0, n, j^k/j!));
%Y A368759 Columns k=0..3 give A038159, A033540(n+1), A053817, A368760.
%Y A368759 Cf. A337085.
%K A368759 nonn,tabl
%O A368759 0,1
%A A368759 _Seiichi Manyama_, Jan 04 2024