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A337085 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.

%I A337085 #26 Jan 04 2024 09:11:06
%S A337085 1,0,2,0,1,5,0,1,4,16,0,1,6,15,65,0,1,10,27,64,326,0,1,18,57,124,325,
%T A337085 1957,0,1,34,135,292,645,1956,13700,0,1,66,345,796,1585,3906,13699,
%U A337085 109601,0,1,130,927,2404,4605,9726,27391,109600,986410,0,1,258,2577,7804,15145,28926,68425,219192,986409,9864101
%N A337085 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.
%H A337085 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%H A337085 Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>
%F A337085 T(0,k) = 0^k and T(n,k) = n^k + n * T(n-1,k) for n>0.
%F A337085 E.g.f. of column k: B_k(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - _Seiichi Manyama_, Jan 04 2024
%e A337085 Square array begins:
%e A337085      1,    0,    0,    0,     0,     0,      0, ...
%e A337085      2,    1,    1,    1,     1,     1,      1, ...
%e A337085      5,    4,    6,   10,    18,    34,     66, ...
%e A337085     16,   15,   27,   57,   135,   345,    927, ...
%e A337085     65,   64,  124,  292,   796,  2404,   7804, ...
%e A337085    326,  325,  645, 1585,  4605, 15145,  54645, ...
%e A337085   1957, 1956, 3906, 9726, 28926, 98646, 374526, ...
%t A337085 T[n_, k_] := n! * Sum[If[j == k == 0, 1, j^k]/j!, {j, 0, n}]; Table[T[k, n-k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Apr 29 2021 *)
%Y A337085 Columns k=0..5 give A000522, A007526, A030297, A337001, A337002, A368719.
%Y A337085 Main diagonal gives A256016.
%Y A337085 Cf. A368724.
%K A337085 nonn,tabl
%O A337085 0,3
%A A337085 _Seiichi Manyama_, Aug 14 2020