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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)*LambertW(-x).
2

%I #10 Nov 09 2017 21:18:42

%S 0,0,1,0,1,2,0,1,4,9,0,1,6,18,64,0,1,8,33,116,625,0,1,10,54,216,1060,

%T 7776,0,1,12,81,388,1865,12702,117649,0,1,14,114,656,3340,21228,

%U 187810,2097152,0,1,16,153,1044,5905,36414,303765,3296120,43046721,0,1,18,198,1576,10100,63480,500374,5222864,66897288,1000000000

%N 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)*LambertW(-x).

%H G. C. Greubel, <a href="/A294411/b294411.txt">Table of n, a(n) for the first 50 antidiagonals, flattened</a>

%F E.g.f. of column k: -exp(k*x)*LambertW(-x).

%e E.g.f. of column k: A_k(x) = x/1! + 2*(k + 1)*x^2/2! + 3*(k^2 + 2*k + 3)*x^3/3! + 4*(k^3 + 3*k^2 + 9*k + 16)*x^4/4! + ...

%e Square array begins:

%e 0, 0, 0, 0, 0, 0, ...

%e 1, 1, 1, 1, 1, 1, ...

%e 2, 4, 6, 8, 10, 12, ...

%e 9, 18, 33, 54, 81, 114, ...

%e 64, 116, 216, 388, 656, 1044, ...

%e 625, 1060, 1895, 3340, 5905, 10100, ...

%t Table[Function[k, n! SeriesCoefficient[-Exp[k x] LambertW[-x], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

%Y Columns k=0..2 give A000169, A277473, A277485.

%Y Main diagonal gives A292633.

%Y Cf. A290824.

%K nonn,tabl

%O 0,6

%A _Ilya Gutkovskiy_, Oct 30 2017