A234289 E.g.f. satisfies: A(x) = 1 + A(x)^2 * Integral 1/A(x) dx.

1, 1, 3, 17, 147, 1729, 25827, 468593, 10012083, 246287521, 6856204803, 213102768977, 7315460977107, 274894137157249, 11223280473993507, 494715928976218673, 23416019742035332083, 1184519963466363339361, 63774753426394808946243, 3641219528568659379843857

Offset

0,3

Comments

Compare to: G(x) = 1 + G(x)^2 * Integral 1/G(x)^2 dx, where G(x) is the e.g.f. of A006351, the number of series-parallel networks with n labeled edges.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

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

E.g.f.: 1 + Series_Reversion( 2*x/(1+x) - log(1+x) ).

E.g.f.: -2/LambertW(-1,-2*exp(x-2)). - Vaclav Kotesovec, Dec 27 2013

E.g.f.: A(x) = C( Integral 1/A(x) dx ), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x), is the Catalan function of A000108. - Paul D. Hanna, May 23 2019

a(n) ~ 2 * n^(n-1) / (exp(n) * (1-log(2))^(n-1/2)). - Vaclav Kotesovec, Dec 27 2013

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 147*x^4/4! + 1729*x^5/5! +...
where A(x)^2 = 1 + 2*x + 8*x^2/2! + 52*x^3/3! + 484*x^4/4! + 5948*x^5/5! +...
Integral 1/A(x) dx = x - x^2/2! - x^3/3! - 5*x^4/4! - 41*x^5/5! - 469*x^6/6! +...
Further,
Series_Reversion(Integral 1/A(x) dx) = x + x^2/2 + 2*x^3/3 + 5*x^4/4 + 14*x^5/5 + 42*x^6/6 + 132*x^7/7 +...+ A000108(n-1)*x^n/n +...
where A000108(n) = binomial(2*n,n)/(n+1).

Maple

seq(n! * coeff(series(-2/LambertW(-1, -2*exp(x-2)), x, n+1), x, n), n = 0..10) # Vaclav Kotesovec, Dec 27 2013

Mathematica

CoefficientList[1 + InverseSeries[Series[2*x/(1+x) - Log[1+x], {x, 0, 20}], x], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 27 2013 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+A^2*intformal(1/(A+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Explicit formula using Catalan function C(x) = 1 + x*C(x)^2: */
{a(n)=local(C=(1-sqrt(1-4*x+x^2*O(x^n)))/(2*x), A=1); A=1/deriv(serreverse(intformal(C))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Explicit formula: 1 + Series_Reversion(2*x/(1+x) - log(1+x)): */
{a(n)=local(A=1, X=x+x^2*O(x^n)); A=1+serreverse(2*X/(1+X)-log(1+X)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A000108, A234290.

Sequence in context: A138013 A052807 A080253 * A009813 A319946 A213507

Adjacent sequences: A234286 A234287 A234288 * A234290 A234291 A234292

Keyword

nonn

Author

Paul D. Hanna, Dec 22 2013

Status

approved