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A335975 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*(exp(x) - 1) + x).

%I A335975 #19 Jul 06 2020 10:41:00
%S A335975 1,1,1,1,2,1,1,3,5,1,1,4,11,15,1,1,5,19,47,52,1,1,6,29,103,227,203,1,
%T A335975 1,7,41,189,622,1215,877,1,1,8,55,311,1357,4117,7107,4140,1,1,9,71,
%U A335975 475,2576,10589,29521,44959,21147,1,1,10,89,687,4447,23031,88909,227290,305091,115975,1
%N A335975 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*(exp(x) - 1) + x).
%H A335975 Seiichi Manyama, <a href="/A335975/b335975.txt">Antidiagonals n = 0..139, flattened</a>
%F A335975 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 A335975 T(n,k) = exp(-k) * Sum_{j>=0} (j + 1)^n * k^j / j!.
%e A335975 Square array begins:
%e A335975   1,   1,    1,     1,     1,      1,      1, ...
%e A335975   1,   2,    3,     4,     5,      6,      7, ...
%e A335975   1,   5,   11,    19,    29,     41,     55, ...
%e A335975   1,  15,   47,   103,   189,    311,    475, ...
%e A335975   1,  52,  227,   622,  1357,   2576,   4447, ...
%e A335975   1, 203, 1215,  4117, 10589,  23031,  44683, ...
%e A335975   1, 877, 7107, 29521, 88909, 220341, 478207, ...
%t A335975 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, 10}, {k, n, 0, -1}] // Flatten (* _Amiram Eldar_, Jul 03 2020 *)
%Y A335975 Columns k=0-4 give: A000012, A000110(n+1), A035009(n+1), A078940, A078945.
%Y A335975 Main diagonal gives A334240.
%Y A335975 Cf. A292860, A335977.
%K A335975 nonn,tabl
%O A335975 0,5
%A A335975 _Seiichi Manyama_, Jul 03 2020