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A294411
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
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, 7776, 0, 1, 12, 81, 388, 1865, 12702, 117649, 0, 1, 14, 114, 656, 3340, 21228, 187810, 2097152, 0, 1, 16, 153, 1044, 5905, 36414, 303765, 3296120, 43046721, 0, 1, 18, 198, 1576, 10100, 63480, 500374, 5222864, 66897288, 1000000000
OFFSET
0,6
FORMULA
E.g.f. of column k: -exp(k*x)*LambertW(-x).
EXAMPLE
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! + ...
Square array begins:
0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, ...
2, 4, 6, 8, 10, 12, ...
9, 18, 33, 54, 81, 114, ...
64, 116, 216, 388, 656, 1044, ...
625, 1060, 1895, 3340, 5905, 10100, ...
MATHEMATICA
Table[Function[k, n! SeriesCoefficient[-Exp[k x] LambertW[-x], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
CROSSREFS
Columns k=0..2 give A000169, A277473, A277485.
Main diagonal gives A292633.
Cf. A290824.
Sequence in context: A351640 A173003 A335461 * A274390 A244128 A016584
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Oct 30 2017
STATUS
approved