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A300014 E.g.f.: (1 + LambertW(-x))^(x/LambertW(-x)). 2
1, 1, 2, 9, 70, 760, 10451, 173159, 3350796, 74100408, 1842574557, 50873252287, 1543925883754, 51078021158476, 1829361243438535, 70512079627989757, 2910210706666701048, 128046344157824920272, 5982882723357716484777, 295846337542679153759691, 15435034787110130135765446, 847314120718756196303796884, 48820553176413784195229252939, 2945834523040528378700506066289 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Logarithm of e.g.f. yields the e.g.f. of A304866.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..385

FORMULA

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n / n! satisfies:

(1) Sum_{n>=0} (n*x - log(A(x)))^n / n!  =  1.

(2) Sum_{n>=0} (n*x - p*log(A(x)))^n / n!  =  (1 + LambertW(-x))^(p-1).

(3) Sum_{n>=0} ((n + m)*x - log(A(x)))^n / n!  =  ( LambertW(-x)/(-x) )^m.

(4) Sum_{n>=0} ((n + m)*x - p*log(A(x)))^n / n!  =  ( LambertW(-x)/(-x) )^m * (1 + LambertW(-x))^(p-1).

a(n) ~ 2^(1/2 - exp(-1)/2) * sqrt(Pi) * n^(n + exp(-1)/2 - 1/2) / Gamma(exp(-1)/2) * (1 + sqrt(2)*log(n)*Gamma(exp(-1)/2 + 1) / (sqrt(n)*Gamma(exp(-1)/2 - 1/2))). - Vaclav Kotesovec, Sep 01 2020

EXAMPLE

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 70*x^4/4! + 760*x^5/5! + 10451*x^6/6! + 173159*x^7/7! + 3350796*x^8/8! + 74100408*x^9/9! + 1842574557*x^10/10! + ...

RELATED SERIES.

log(A(x)) = x + x^2/2! + 5*x^3/3! + 40*x^4/4! + 434*x^5/5! + 5921*x^6/6! + 97152*x^7/7! + 1861224*x^8/8! + 40757712*x^9/9! + ... + A304866(n)*x^n/n! + ...

such that Sum_{n>=0} (n*x - log(A(x)))^n / n!  =  1.

MATHEMATICA

nmax = 20; CoefficientList[Series[(1 + LambertW[-x])^(x/LambertW[-x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 01 2020 *)

PROG

(PARI) {a(n) = my(W = serreverse(-x*exp(x +x*O(x^n)))); n!*polcoeff( (1 + W)^(x/W), n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A304866.

Sequence in context: A296718 A201862 A167016 * A108522 A014500 A101482

Adjacent sequences:  A300011 A300012 A300013 * A300015 A300016 A300017

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 18 2018

STATUS

approved

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Last modified September 20 12:05 EDT 2021. Contains 347586 sequences. (Running on oeis4.)