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A334165 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(-1/k) * Sum_{j>=0} (k*j + 1)^n / (k^j * j!).

%I A334165 #4 Apr 17 2020 07:20:00
%S A334165 1,1,2,1,2,5,1,2,6,15,1,2,7,24,52,1,2,8,35,116,203,1,2,9,48,214,648,
%T A334165 877,1,2,10,63,352,1523,4088,4140,1,2,11,80,536,3008,12349,28640,
%U A334165 21147,1,2,12,99,772,5307,29440,112052,219920,115975,1,2,13,120,1066,8648,60389,324096,1120849,1832224,678570
%N A334165 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(-1/k) * Sum_{j>=0} (k*j + 1)^n / (k^j * j!).
%C A334165 Square array of Dowling numbers.
%H A334165 Moussa Benoumhani, <a href="https://doi.org/10.1016/0012-365X(95)00095-E">On Whitney numbers of Dowling lattices</a>, Discrete Math. 159 (1996), no. 1-3, 13-33.
%F A334165 G.f. of column k: (1/(1 - x)) * Sum_{j>=0} (x/(1 - x))^j / Product_{i=1..j} (1 - k*i*x/(1 - x)).
%F A334165 E.g.f. of column k: exp(x + (exp(k*x) - 1) / k).
%e A334165 Square array begins:
%e A334165     1,    1,     1,     1,     1,     1,  ...
%e A334165     2,    2,     2,     2,     2,     2,  ...
%e A334165     5,    6,     7,     8,     9,    10,  ...
%e A334165    15,   24,    35,    48,    63,    80,  ...
%e A334165    52,  116,   214,   352,   536,   772,  ...
%e A334165   203,  648,  1523,  3008,  5307,  8648,  ...
%t A334165 Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[(x/(1 - x))^j/Product[(1 - k i x/(1 - x)), {i, 1, j}], {j, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
%t A334165 Table[Function[k, n! SeriesCoefficient[Exp[x + (Exp[k x] - 1)/k], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
%Y A334165 Columns k=1..10 give A000110 (for n > 0), A007405, A003575, A003576, A003577, A003578, A003579, A003580, A003581, A003582.
%Y A334165 Cf. A241578, A241579, A334162 (diagonal).
%K A334165 nonn,tabl
%O A334165 0,3
%A A334165 _Ilya Gutkovskiy_, Apr 17 2020