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A335977 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).

%I A335977 #30 Jul 07 2020 08:56:18
%S A335977 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 A335977 11,5,-8,-13,9,1,1,-6,19,-9,-43,-65,-89,-50,1,1,-7,29,-41,-74,-27,37,
%U A335977 -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 A335977 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 A335977 Seiichi Manyama, <a href="/A335977/b335977.txt">Antidiagonals n = 0..139, flattened</a>
%F A335977 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 A335977 T(n,k) = exp(k) * Sum_{j>=0} (j + 1)^n * (-k)^j / j!.
%e A335977 Square array begins:
%e A335977   1,  1,   1,   1,   1,   1,    1, ...
%e A335977   1,  0,  -1,  -2,  -3,  -4,   -5, ...
%e A335977   1, -1,  -1,   1,   5,  11,   19, ...
%e A335977   1, -1,   3,   7,   5,  -9,  -41, ...
%e A335977   1,  2,   7,  -8, -43, -74,  -53, ...
%e A335977   1,  9, -13, -65, -27, 221,  679, ...
%e A335977   1,  9, -89,  37, 597, 961, -341, ...
%t A335977 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 A335977 Columns k=0-4 give: A000012, A293037, A309775, A320432, A320433.
%Y A335977 Main diagonal gives A334241.
%Y A335977 Cf. A292861, A335975.
%K A335977 sign,tabl,look
%O A335977 0,12
%A A335977 _Seiichi Manyama_, Jul 03 2020