A217061 Expansion of e.g.f. exp(A006351(x)).

1, 1, 3, 15, 109, 1053, 12767, 186763, 3204313, 63128665, 1404963387, 34867190823, 954800951749, 28600649870133, 930325531322519, 32658109219519843, 1230609634110271921, 49545182501048868145, 2122562841050605554291, 96411483206025310956735, 4628163318874435745244445

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = sum(m=1..n, (sum(k=0..n-m, (n+k-1)!*sum(j=0..k, 1/(k-j)!*sum(l=0..j, (2^(j-l)*(-1)^(l+j)*Stirling1(n-m-l+j,j-l))/(l!*(n-m-l+j)!)))))/(m-1)!), n>0, a(0)=1.

From Vaclav Kotesovec, Aug 04 2014: (Start)

E.g.f.: 4*LambertW(-exp((x-1)/2)/2)^2 / exp(x).

a(n) ~ sqrt(2) * n^(n-1) / (exp(n-1) * (2*log(2)-1)^(n-1/2)). (End)

Mathematica

CoefficientList[Series[4*ProductLog[-E^((x-1)/2)/2]^2/E^x, {x, 0, 15}], x]*Range[0, 15]! (* Vaclav Kotesovec, Aug 04 2014 *)

Prog

(Maxima)
a(n):=(sum((m*sum((n+k-1)!*sum(1/(k-j)!*sum((2^(j-l)*(-1)^(l+j)*stirling1(n-m-l+j, j-l))/(l!*(n-m-l+j)!), l, 0, j), j, 0, k), k, 0, n-m))/m!, m, 1, n));
(PARI) my(x='x+O('x^20)); apply(round, Vec(serlaplace(4*lambertw(-exp((x-1)/2)/2)^2 / exp(x)))) \\ Michel Marcus, Jan 27 2025

Crossrefs

Cf. A006351, A379458.

Sequence in context: A153305 A380080 A110328 * A054201 A090355 A083483

Adjacent sequences: A217058 A217059 A217060 * A217062 A217063 A217064

Keyword

nonn,changed

Author

Vladimir Kruchinin, Sep 26 2012

Extensions

More terms from Michel Marcus, Jan 27 2025

Status

approved