A003725 Expansion of e.g.f. exp( x * exp(-x) ). (Formerly M1911)

1, 1, -1, -2, 9, -4, -95, 414, 49, -10088, 55521, -13870, -2024759, 15787188, -28612415, -616876274, 7476967905, -32522642896, -209513308607, 4924388011050, -38993940088199, 11731860520780, 3807154270837281, -52018152493218010, 278413297030360273, 2454092710416045576

Offset

0,4

Comments

a(n) alternates in parity (follows from Lucas' theorem) and is divisible by n-1 (follows from Abel's identity). - Aidan J. Ruck, Oct 04 2025

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..557

Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3.

Formula

a(n) = Sum_{k=0..n} (-k)^(n-k)*binomial(n, k). - Vladeta Jovovic, Mar 15 2003

First column of A215652. - Peter Bala, Sep 14 2012

G.f.: Sum_{k>=0} x^k/(1 + k*x)^(k+1). - Ilya Gutkovskiy, Jun 25 2018

Mathematica

With[{nn=30}, CoefficientList[Series[Exp[x Exp[-x]], {x, 0, nn}], x]Range[0, nn]!] (* Harvey P. Dale, Oct 20 2011 *)

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(exp(-x) * x))) \\ Charles R Greathouse IV, Sep 26 2017

Crossrefs

Cf. A000248, A069856, A072034, A215652.

Cf. this sequence (k=1), A292909 (k=2), A292910 (k=3), A292912 (k=4).

Sequence in context: A127145 A210423 A292952 * A328619 A365108 A076930

Adjacent sequences: A003722 A003723 A003724 * A003726 A003727 A003728

Keyword

sign,easy

Author

R. H. Hardin

Status

approved