A141142 Numerators of power series arising in random graphs.

-1, 2, -4, 44, -104, 40, -7648, 2848, -31712, 23429344, -89072576, 1441952704, -893393408, 9352282112, -11547336704, 314833934543872, -886909037097472, 8407858707080704, -3185585650598165504, 476968653230369792, -4605749416183789568, 11898401315301146359808, -282034907680992595910656, 1032668724971184398336, -2345904699036953797820416

Offset

1,2

Comments

Denominators are A141143. Series given in Bouchard and Marino 2.31, p. 5, with citations to earlier literature.

Links

Table of n, a(n) for n=1..25.

Vincent Bouchard and Marcos Marino, Hurwitz numbers, matrix models and enumerative geometry, arXiv:0709.1458 [math.AG], 2007-2008.

Formula

Power series s(z) defined by (1+z)*exp(-z) = (1 + s(z))*exp(-s(z)).

s(z) = -1 - LambertW(-(1+z)*exp(-z-1)). - Max Alekseyev, Aug 17 2013

Example

s(z) = -z + (2/3)*z^2 - (4/9)*z^3 + (44/135)*z^4 - (104/405)*z^5 + (40/189)*z^6 - (7648/42525)*z^7 + O(z^8).

Maple

assume(z>0); series( -1 - LambertW(-(1+z)*exp(-z-1)), z, 20 );

Mathematica

Assuming[z>0, Series[-1 - ProductLog[-(1+z) Exp[-1-z]], {z, 0, 25}]] // CoefficientList[#, z]& // Numerator (* Jean-François Alcover, Aug 13 2018, after Max Alekseyev *)

Crossrefs

Cf. A141143.

Sequence in context: A359876 A059794 A116611 * A353797 A007596 A050588

Adjacent sequences: A141139 A141140 A141141 * A141143 A141144 A141145

Keyword

frac,sign

Author

Jonathan Vos Post, Jun 10 2008

Extensions

Terms a(8) onward from Max Alekseyev, Aug 17 2013

Status

approved