A144012 E.g.f. satisfies: A'(x) = 1 + x*A(x)^2 where A(0) = 1.

1, 1, 1, 4, 12, 56, 310, 1872, 13804, 110368, 990792, 9816560, 105392056, 1231910208, 15473322592, 208287327136, 2992281160320, 45647837225984, 737580584547424, 12578608722516480, 225799744451927104

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

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

Let r be the radius of convergence of e.g.f. A(x), then: a(n)/n! ~ r^(n+2) where r=0.89757966985304971385345783421642045642527022484..., A(-r)=0.206876159989240..., A'(-r)=0.961585613659124...

r is the root of the equation 2*r^2*Hypergeometric0F1[1/3,-1/(9*r^3)] = Hypergeometric0F1[5/3,-1/(9*r^3)]. - Vaclav Kotesovec, Feb 23 2014

Example

E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 12*x^4/4! + 56*x^5/5! +...
A(x)^2 = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 62*x^4/4! + 312*x^5/5! +...
x*A(x)^2 = x + 4*x^2/2! + 12*x^3/3! + 56*x^4/4! + 310*x^5/5! +...
A'(x) = 1 + x + 4*x^2/2! + 12*x^3/3! + 56*x^4/4! + 310*x^5/5! +...

Mathematica

CoefficientList[Series[-2*(Hypergeometric0F1[2/3, -x^3/9] + x*Hypergeometric0F1[4/3, -x^3/9]) / (-2*Hypergeometric0F1[1/3, -x^3/9] + x^2*Hypergeometric0F1[5/3, -x^3/9]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Dec 21 2013 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1+x*(A+x*O(x^n))^2)); n!*polcoeff(A, n)}

Crossrefs

Cf. A144013, A144014.

Sequence in context: A243785 A019266 A009114 * A197924 A065125 A208940

Adjacent sequences: A144009 A144010 A144011 * A144013 A144014 A144015

Keyword

nonn

Author

Paul D. Hanna, Sep 10 2008

Status

approved