A234239 E.g.f. satisfies: A(x) = exp( x + Integral Integral A(x)^3 dx dx ).

1, 1, 2, 7, 34, 209, 1558, 13663, 137786, 1570681, 19970182, 280168967, 4299033994, 71619894529, 1287342696278, 24832567401103, 511673425673626, 11215927371237161, 260604889591097062, 6397958871977787127, 165486967875852965354, 4498061784752926891249, 128176486634710543231798

Offset

0,3

Comments

Compare to: F(x) = exp(x + Integral Integral F(x) dx dx) holds when F(x) = 1/(1-sin(x)).

Compare to: G(x) = exp(x + Integral Integral G(x)^2 dx dx) holds when G(x) = 1/(1-x).

Links

Table of n, a(n) for n=0..22.

Formula

E.g.f.: 1/(2*cosh(sqrt(3)*x) - sqrt(3)*sinh(sqrt(3)*x) - 1)^(1/3). - Vaclav Kotesovec, Jan 05 2014

a(n) ~ n! * 2^(1/3) * 3^(n/2) / (GAMMA(2/3) * n^(1/3) * (log(2+sqrt(3)))^(n+2/3)). - Vaclav Kotesovec, Jan 05 2014

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 34*x^4/4! + 209*x^5/5! +...
where
A(x)^3 = 1 + 3*x + 12*x^2/2! + 63*x^3/3! + 414*x^4/4! + 3267*x^5/5! +...
log(A(x)) = x + x^2/2! + 3*x^3/3! + 12*x^4/4! + 63*x^5/5! + 414*x^6/6! + 3267*x^7/7! +...

Mathematica

CoefficientList[Series[(1/(-1 + 2*Cosh[Sqrt[3]*x] - Sqrt[3]*Sinh[Sqrt[3]*x]))^(1/3), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 05 2014 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(x+intformal(intformal(A^3+x*O(x^n))))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Sequence in context: A145845 A355292 A002720 * A249833 A111539 A337000

Adjacent sequences: A234236 A234237 A234238 * A234240 A234241 A234242

Keyword

nonn

Author

Paul D. Hanna, Dec 21 2013

Status

approved