A143922 E.g.f. A(x) satisfies: A(x) = 1 + x*exp(Integral A(x)^2 dx).

1, 1, 2, 9, 52, 395, 3666, 40257, 510600, 7343523, 118093310, 2099660497, 40896662124, 866008634907, 19808285169834, 486698217317985, 12784410332144656, 357512156423101427, 10604399352362692182

Offset

0,3

Comments

Compare definition of e.g.f. A(x) to the trivial statement:

if F(x) = 1/(1-x) then F(x) = 1 + x*exp(Integral F(x) dx).

Here Integral F(x) dx does not include the constant of integration.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Formula

E.g.f. derivative: A'(x) = [1 + x*A(x)^2]*(A(x) - 1)/x.

a(n) ~ n^n / (exp(n) * r^(n+1/2)), where r = 0.58963282569434540653295100228290669896338789564481715119... - Vaclav Kotesovec, Feb 20 2014

Example

E.g.f. A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 395*x^5/5! +...
A(x)^2 = 1 + 2*x + 6*x^2/2! + 30*x^3/3! + 200*x^4/4! + 1670*x^5/5! +...
Let L(x) = Integral A(x)^2 dx where A(x) = 1 + x*exp(L(x)), then
L(x) = x + 2*x^2/2! + 6*x^3/3! + 30*x^4/4! + 200*x^5/5! +...
exp(L(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 79*x^4/4! + 611*x^5/5! +...

Mathematica

a = ConstantArray[0, 20]; a[[1]]=1; a[[2]]=1; Do[a[[n+1]] = (-n! * Sum[a[[i+1]] * a[[n-i]]/i!/(n-i-1)!, {i, 0, n-1}] + n! * Sum[a[[k+1]]/k! * Sum[a[[i+1]]*a[[n-k-i]]/i!/(n-k-i-1)!, {i, 0, n-1}], {k, 0, n-1}])/(n-1), {n, 2, 19}]; a  (* Vaclav Kotesovec, Feb 20 2014 *)

Prog

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

Crossrefs

Cf. A143923, A143924.

Sequence in context: A052882 A248440 A330200 * A305304 A369090 A110322

Adjacent sequences: A143919 A143920 A143921 * A143923 A143924 A143925

Keyword

nonn

Author

Paul D. Hanna, Sep 06 2008

Status

approved