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A290824 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(1 + LambertW(-x)).

%I A290824 #16 Jul 17 2024 15:56:12
%S A290824 1,1,1,1,2,4,1,3,7,27,1,4,12,43,256,1,5,19,71,393,3125,1,6,28,117,616,
%T A290824 4721,46656,1,7,39,187,985,7197,69853,823543,1,8,52,287,1584,11123,
%U A290824 105052,1225757,16777216,1,9,67,423,2521,17429,159093,1829291,24866481,387420489,1,10,84,601,3928,27525,243256,2740111,36922928,572410513,10000000000
%N A290824 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(1 + LambertW(-x)).
%C A290824 A(n,k) is the k-th binomial transform of A000312 evaluated at n.
%H A290824 G. C. Greubel, <a href="/A290824/b290824.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A290824 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A290824 E.g.f. of column k: exp(k*x)/(1 + LambertW(-x)).
%F A290824 A(n,k) = Sum_{j=0..n} binomial(n,j)*k^(n-j)*j^j. - _Fabian Pereyra_, Jul 16 2024
%e A290824 E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 4)*x^2/2! + (k^3 + 3*k^2 + 12*k + 27)*x^3/3! + (k^4 + 4*k^3 + 24*k^2 + 108*k + 256)*x^4/4! + ...
%e A290824 Square array begins:
%e A290824      1,     1,     1,     1,     1,     1, ...
%e A290824      1,     2,     3,     4,     5,     6, ...
%e A290824      4,     7,    12,    19,    28,    39, ...
%e A290824     27,    43,    71,   117,   187,   287, ...
%e A290824    256,   393,   616,   985,  1584,  2521, ...
%e A290824   3125,  4721,  7197, 11123, 17429, 27525, ...
%t A290824 Table[Function[k, n!*SeriesCoefficient[Exp[k x]/(1 + LambertW[-x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten (* _G. C. Greubel_, Nov 09 2017 *)
%Y A290824 Columns k=0..2 give A000312, A086331, A277457.
%Y A290824 Main diagonal gives A290840.
%Y A290824 Cf. A080955, A081576, A271025.
%K A290824 nonn,tabl
%O A290824 0,5
%A A290824 _Ilya Gutkovskiy_, Aug 11 2017