A234293 E.g.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^3 dx.

1, 1, 5, 57, 1053, 27057, 891765, 35883369, 1705399821, 93486709089, 5806695485925, 403035074738073, 30915209058366717, 2597006055407919633, 237114260894842246485, 23379981536098259658441, 2475981772053312305191149, 280285314108547303508788161, 33774891856931450901116286405

Offset

0,3

Links

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

Formula

E.g.f.: 1 + Series_Reversion( 4*x/(1+x) - 3*log(1+x) ).

E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x)^3 dx ) )^(1/3), where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

a(n) ~ n! * (2/3)^(3/2) / (sqrt(Pi) * n^(3/2) * (1-6*log(2)+3*log(3))^(n-1/2)). - Vaclav Kotesovec, Dec 26 2013

E.g.f.: -4/(3*LambertW(-1,-4/3*exp((x-4)/3))). - Vaclav Kotesovec, Dec 26 2013

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 57*x^3/3! + 1053*x^4/4! + 27057*x^5/5! +...
where A(4*x/(1+x) - 3*log(1+x)) = 1+x.
Related series:
A(x)^3 = 1 + 3*x + 21*x^2/2! + 267*x^3/3! + 5157*x^4/4! + 135531*x^5/5! +...
A(x)^4 = 1 + 4*x + 32*x^2/2! + 432*x^3/3! + 8592*x^4/4! + 229488*x^5/5! +...
1/A(x)^3 = 1 - 3*x - 3*x^2/2! - 51*x^3/3! - 963*x^4/4! - 25011*x^5/5! +...
The series reversion of A(x)-1 begins:
4*x/(1+x) - 3*log(1+x) = x - 5*x^2/2 + 9*x^3/3 - 13*x^4/4 + 17*x^5/5 - 21*x^6/6 +...

Maple

seq(n! * coeff(series(-4/(3*LambertW(-1, -4/3*exp((x-4)/3))), x, n+1), x, n), n = 0..10) # Vaclav Kotesovec, Dec 26 2013

Mathematica

CoefficientList[1 + InverseSeries[Series[4*x/(1+x)-3*Log[1+x], {x, 0, 20}], x], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 26 2013 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+A^4*intformal(1/(A^3+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=local(A=1, X=x+x^2*O(x^n)); A=1+serreverse(4*X/(1+X)-3*log(1+X)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A234292, A234294, A234295.

Sequence in context: A336286 A155539 A285067 * A244288 A009480 A009481

Adjacent sequences: A234290 A234291 A234292 * A234294 A234295 A234296

Keyword

nonn

Author

Paul D. Hanna, Dec 25 2013

Status

approved