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A257476 E.g.f. satisfies: A'(x) = exp(A(x)) / (1 - A(x))^2. 1
1, 3, 20, 208, 2944, 52748, 1143876, 29120292, 851236008, 28097170632, 1033540588992, 41922955409520, 1858913275469808, 89451843240275280, 4642580506992107328, 258501607230300771072, 15370815613407241773312, 972082967929423498417344, 65152216036788689542930752, 4613084984991245705602298688 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Compare to: W'(x) = exp(W(x)) / (1 - W(x)), which holds when W(x) = -LambertW(-x).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..330

V. Kotesovec, Asymptotic of implicit functions if Fww = 0

FORMULA

E.g.f.: Series_Reversion( Sum_{n>=1} -(n^2-n+1) * (-x)^n/n! ).

a(n) ~ GAMMA(1/3) * n^(n-5/6) / (3^(1/6) * sqrt(2*Pi) * (exp(1)-2)^(n-1/3)). - Vaclav Kotesovec, Apr 26 2015

EXAMPLE

E.g.f.: A(x) = x + 3*x^2/2! + 20*x^3/3! + 208*x^4/4! + 2944*x^5/5! +...

where

exp(A(x))/(1-A(x))^2 = 1 + 3*x + 20*x^2/2! + 208*x^3/3! + 2944*x^4/4! +...

The e.g.f. also satisfies:

x = Sum_{n>=1} (-1)^(n-1) * (n^2-n+1) * A(x)^n/n! , or

x = 1 - exp(-A(x)) * (1 + A(x)^2)  [after Vaclav Kotesovec];

explicitly,

x = A(x)*x - 3*A(x)^2*x^2/2! + 7*A(x)^3*x^3/3! - 13*A(x)^4*x^4/4! + 21*A(x)^5*x^5/5! - 31*A(x)^6*x^6/6! +...

MATHEMATICA

Rest[CoefficientList[InverseSeries[Series[1-E^(-x)*(1+x^2), {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Apr 26 2015 *)

PROG

(PARI) {a(n)=local(A=x); for(i=1, n, A = intformal((1-A)^-2*(exp(A+x*O(x^n)))) ); n!*polcoeff(A, n)}

for(n=1, 30, print1(a(n), ", "))

(PARI) {a(n)=local(A=x); A = serreverse(sum(m=1, n, (-1)^(m-1)*(m^2-m+1)*x^m/m!) +x*O(x^n)); n!*polcoeff(A, n)}

for(n=1, 30, print1(a(n), ", "))

CROSSREFS

Sequence in context: A052590 A081209 A196560 * A218673 A230478 A014068

Adjacent sequences:  A257473 A257474 A257475 * A257477 A257478 A257479

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 25 2015

STATUS

approved

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Last modified July 12 18:03 EDT 2020. Contains 335666 sequences. (Running on oeis4.)