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%I #30 Jul 07 2020 08:56:18
%S 1,1,1,1,0,1,1,-1,-1,1,1,-2,-1,-1,1,1,-3,1,3,2,1,1,-4,5,7,7,9,1,1,-5,
%T 11,5,-8,-13,9,1,1,-6,19,-9,-43,-65,-89,-50,1,1,-7,29,-41,-74,-27,37,
%U -45,-267,1,1,-8,41,-97,-53,221,597,1024,1191,-413,1,1,-9,55,-183,92,679,961,805,1351,4723,2180,1
%N Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(1 - exp(x)) + x).
%H Seiichi Manyama, <a href="/A335977/b335977.txt">Antidiagonals n = 0..139, flattened</a>
%F T(0,k) = 1 and T(n,k) = T(n-1,k) - k * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.
%F T(n,k) = exp(k) * Sum_{j>=0} (j + 1)^n * (-k)^j / j!.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 0, -1, -2, -3, -4, -5, ...
%e 1, -1, -1, 1, 5, 11, 19, ...
%e 1, -1, 3, 7, 5, -9, -41, ...
%e 1, 2, 7, -8, -43, -74, -53, ...
%e 1, 9, -13, -65, -27, 221, 679, ...
%e 1, 9, -89, 37, 597, 961, -341, ...
%t T[0, k_] := 1; T[n_, k_] := T[n - 1, k] - k * Sum[T[j, k] * Binomial[n - 1, j], {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* _Amiram Eldar_, Jul 03 2020 *)
%Y Columns k=0-4 give: A000012, A293037, A309775, A320432, A320433.
%Y Main diagonal gives A334241.
%Y Cf. A292861, A335975.
%K sign,tabl,look
%O 0,12
%A _Seiichi Manyama_, Jul 03 2020
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