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Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.
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%I #26 Jan 04 2024 09:11:06

%S 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 1957,0,1,34,135,292,645,1956,13700,0,1,66,345,796,1585,3906,13699,

%U 109601,0,1,130,927,2404,4605,9726,27391,109600,986410,0,1,258,2577,7804,15145,28926,68425,219192,986409,9864101

%N 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>

%F T(0,k) = 0^k and T(n,k) = n^k + n * T(n-1,k) for n>0.

%F 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 Square array begins:

%e 1, 0, 0, 0, 0, 0, 0, ...

%e 2, 1, 1, 1, 1, 1, 1, ...

%e 5, 4, 6, 10, 18, 34, 66, ...

%e 16, 15, 27, 57, 135, 345, 927, ...

%e 65, 64, 124, 292, 796, 2404, 7804, ...

%e 326, 325, 645, 1585, 4605, 15145, 54645, ...

%e 1957, 1956, 3906, 9726, 28926, 98646, 374526, ...

%t 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 Columns k=0..5 give A000522, A007526, A030297, A337001, A337002, A368719.

%Y Main diagonal gives A256016.

%Y Cf. A368724.

%K nonn,tabl

%O 0,3

%A _Seiichi Manyama_, Aug 14 2020