A234294 E.g.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^4 dx.

1, 1, 4, 40, 664, 15424, 460576, 16808320, 724904896, 36072438016, 2034328297984, 128223244372480, 8932539799788544, 681536817951791104, 56521548341146402816, 5062454448656689500160, 487013865350356256137216, 50082306316236214342844416, 5482502331779770770018893824

Offset

0,3

Links

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

Formula

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

E.g.f.: -4/3*LambertW(-3/4*exp((x-3)/4)).

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

E.g.f.: 1 / sqrt( d/dx Series_Reversion( Integral (1+2*x*C(2*x))^2/(1+x)^2 dx ) ), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x), is the Catalan function of A000108.

O.g.f.: 1 + x/(1-x - 3*x/(1-2*x - 3*2*x/(1-3*x - 3*3*x/(1-4*x - 3*4*x/(1-...))))), a continued fraction.

a(n) ~ 2 * n^(n-1) / (3 * exp(n) * (8*log(2)-4*log(3)-1)^(n-1/2)). - Vaclav Kotesovec, Dec 26 2013

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 40*x^3/3! + 664*x^4/4! + 15424*x^5/5! +...
where A(4*log(1+x) - 3*x) = 1+x.
Related series:
A(x)^4 = 1 + 4*x + 28*x^2/2! + 328*x^3/3! + 5752*x^4/4! + 137056*x^5/5! +...
1/A(x)^4 = 1 - 4*x + 4*x^2/2! - 40*x^3/3! - 536*x^4/4! - 13216*x^5/5! +...
(d/dx Series_Reversion(Integral 1/A(x)^4 dx))^(1/4) begins:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 +...+ A002293(n)*x^n +...
where G(x) = 1 + x*G(x)^4.

Mathematica

CoefficientList[1 + InverseSeries[Series[4*Log[1+x]-3*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^4+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*log(1+X) - 3*X); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* O.g.f. continued fraction: */
{a(n)=local(CF=1+x*O(x)); for(k=0, n, CF=1-(n-k+1)*x-3*(n-k+1)*x/CF); polcoeff(1+x/CF, n, x)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A006351, A058562, A234295.

Sequence in context: A227055 A343446 A205671 * A181088 A005431 A153849

Adjacent sequences: A234291 A234292 A234293 * A234295 A234296 A234297

Keyword

nonn

Author

Paul D. Hanna, Dec 24 2013

Status

approved