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).

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

Links

G. C. Greubel, Table of n, a(n) for the first 50 antidiagonals, flattened

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: A173003 A378236 A335461 * A274390 A244128 A016584

Adjacent sequences: A294408 A294409 A294410 * A294412 A294413 A294414

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Oct 30 2017

Status

approved