A143920 E.g.f. satisfies: A(x) = 1 + x*exp(2*Integral A(x) dx).

1, 1, 4, 18, 112, 880, 8256, 90384, 1131264, 15927552, 249164800, 4287669760, 80490393600, 1636924403712, 35850727342080, 841260590499840, 21056773882052608, 559992309313503232, 15768699458743959552

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: A330353 A000986 A364623 * A233534 A113356 A062805

Adjacent sequences: A143917 A143918 A143919 * A143921 A143922 A143923

Keyword

nonn

Author

Paul D. Hanna, Sep 05 2008

Status

approved