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)).

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, 4721, 46656, 1, 7, 39, 187, 985, 7197, 69853, 823543, 1, 8, 52, 287, 1584, 11123, 105052, 1225757, 16777216, 1, 9, 67, 423, 2521, 17429, 159093, 1829291, 24866481, 387420489, 1, 10, 84, 601, 3928, 27525, 243256, 2740111, 36922928, 572410513, 10000000000

Offset

0,5

Comments

A(n,k) is the k-th binomial transform of A000312 evaluated at n.

Links

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

N. J. A. Sloane, Transforms

Formula

E.g.f. of column k: exp(k*x)/(1 + LambertW(-x)).

Example

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! + ...
Square array begins:
     1,     1,     1,     1,     1,     1, ...
     1,     2,     3,     4,     5,     6, ...
     4,     7,    12,    19,    28,    39, ...
    27,    43,    71,   117,   187,   287, ...
   256,   393,   616,   985,  1584,  2521, ...
  3125,  4721,  7197, 11123, 17429, 27525, ...

Mathematica

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 *)

Crossrefs

Columns k=0..2 give A000312, A086331, A277457.

Main diagonal gives A290840.

Cf. A080955, A081576, A271025.

Sequence in context: A370020 A338530 A248212 * A272977 A230505 A366062

Adjacent sequences: A290821 A290822 A290823 * A290825 A290826 A290827

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Aug 11 2017

Status

approved