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A368724 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} (-1)^(n-j) * j^k/j!.

%I A368724 #13 Jan 04 2024 08:56:55
%S A368724 1,0,0,0,1,1,0,1,0,-2,0,1,2,3,9,0,1,6,3,-8,-44,0,1,14,9,4,45,265,0,1,
%T A368724 30,39,28,5,-264,-1854,0,1,62,153,100,-15,6,1855,14833,0,1,126,543,
%U A368724 412,125,306,7,-14832,-133496,0,1,254,1809,1924,1065,546,-1799,8,133497,1334961
%N A368724 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} (-1)^(n-j) * j^k/j!.
%H A368724 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%H A368724 Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>
%F A368724 T(0,k) = 0^k and T(n,k) = n^k - n * T(n-1,k) for n>0.
%F A368724 E.g.f. of column k: B_k(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.
%e A368724 Square array begins:
%e A368724     1,    0, 0,   0,   0,    0,     0, ...
%e A368724     0,    1, 1,   1,   1,    1,     1, ...
%e A368724     1,    0, 2,   6,  14,   30,    62, ...
%e A368724    -2,    3, 3,   9,  39,  153,   543, ...
%e A368724     9,   -8, 4,  28, 100,  412,  1924, ...
%e A368724   -44,   45, 5, -15, 125, 1065,  6005, ...
%e A368724   265, -264, 6, 306, 546, 1386, 10626, ...
%o A368724 (PARI) T(n,k) = n!*sum(j=0, n, (-1)^(n-j)*j^k/j!);
%Y A368724 Columns k=0..5 give A182386, (-1)^(n-1) * A000240(n), A001477, A368716, A368717, A368718.
%Y A368724 Main diagonal gives A368725.
%Y A368724 Cf. A337085.
%K A368724 sign,tabl
%O A368724 0,10
%A A368724 _Seiichi Manyama_, Jan 04 2024