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A143920 E.g.f. satisfies: A(x) = 1 + x*exp(2*Integral A(x) dx). 0
1, 1, 4, 18, 112, 880, 8256, 90384, 1131264, 15927552, 249164800, 4287669760, 80490393600, 1636924403712, 35850727342080, 841260590499840, 21056773882052608, 559992309313503232, 15768699458743959552 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare the definition of e.g.f. A(x) to this trivial statement:

if F(x) = 1/(1-2x) then F(x) = 1 + 2*x*exp(2*Integral F(x) dx).

LINKS

Table of n, a(n) for n=0..18.

FORMULA

E.g.f. satisfies: A'(x) = [1 + 2*x*A(x)]*(A(x) - 1)/x where A'(x) = d/dx A(x).

E.g.f.: (1+exp(2*x)) / (1+exp(2*x)*(1-2*x)). - Vaclav Kotesovec, Jan 05 2014

a(n) ~ n! * 2^n / (1 + LambertW(exp(-1)))^(n+1). - Vaclav Kotesovec, Jan 05 2014

E.g.f.: -1/E(0), where E(k)= 4*k-1 + x/(1 - x/(4*k+1 + x/(1 - x/E(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Jan 22 2015

EXAMPLE

E.g.f. A(x) = 1 + x + 4*x^2/2! + 18*x^3/3! + 112*x^4/4! +...

CONVERGENCE AND ASYMPTOTICS.

Let r be the radius of convergence of the power series A(x), then:

a(n)/n! ~ (1/2)/r^(n+1) where

r=0.63923227138053689755467936951149007771973874430987288272658905276...

so that the power series A(x) diverges at x=r.

Note: A(-r) is evaluated as 1/(2r) since Integral A(x) dx is a

convergent alternating series at x=-r having the sum:

Sum_{n>=0} a(n)*(-r)^(n+1)/(n+1)! = log(r - 1/2)/2 - log(r);

however, as N approaches infinity, the N-th partial sum of A(x) at x=-r,

Sum_{n>=0..N} a(n)*(-r)^n/n!, oscillates between 1/(4r) and 3/(4r).

Thus the power series A(x) converges only for |x| < r.

In closed form, r = 1/2 + LambertW(exp(-1))/2. - Vaclav Kotesovec, Jan 05 2014

MATHEMATICA

CoefficientList[Series[(1+E^(2*x))/(1+E^(2*x)*(1-2*x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 05 2014 *)

max = 20; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 4*k-1 + x/(1 - x/(4*k+1 + x/(1 - x/g[k+1]))); gf = -1/g[0]; CoefficientList[Series[gf, {x, 0, max}], x] * Range[0, max]! (* Vaclav Kotesovec, Jan 22 2015, after Sergei N. Gladkovskii *)

PROG

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*exp(2*intformal(A)+x*O(x^n))); n!*polcoeff(A, n)}

CROSSREFS

Sequence in context: A327679 A330353 A000986 * A233534 A113356 A062805

Adjacent sequences:  A143917 A143918 A143919 * A143921 A143922 A143923

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 05 2008

STATUS

approved

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Last modified July 4 16:24 EDT 2020. Contains 335448 sequences. (Running on oeis4.)