A143923 E.g.f. A(x) satisfies: A(x) = 1 + x*exp(Integral A(x)^3 dx).

1, 1, 2, 12, 88, 860, 10392, 149044, 2478752, 46875492, 993291880, 23311581524, 600207989808, 16820818373476, 509711184710840, 16606143020005620, 578830045479469120, 21493718211307208420, 847057099952645864712

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.

Limit n->infinity (a(n)/n!)^(1/n) = 2.274991... - Vaclav Kotesovec, Feb 28 2014

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..176

Formula

E.g.f. derivative: A'(x) = [1 + x*A(x)^3]*(A(x) - 1)/x.

Example

E.g.f. A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 88*x^4/4! + 860*x^5/5! +...
A(x)^3 = 1 + 3*x + 12*x^2/2! + 78*x^3/3! + 696*x^4/4! + 7740*x^5/5! +...
Let L(x) = Integral A(x)^3 dx where A(x) = 1 + x*exp(L(x)), then
L(x) = x + 3*x^2/2! + 12*x^3/3! + 78*x^4/4! + 696*x^5/5! +...
exp(L(x)) = 1 + x + 4*x^2/2! + 22*x^3/3! + 172*x^4/4! + 1732*x^5/5! +...

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*exp(intformal(A^3))); n!*polcoeff(A, n)}

Crossrefs

Cf. A143922, A143924.

Sequence in context: A059435 A192621 A372090 * A079858 A224152 A174356

Adjacent sequences: A143920 A143921 A143922 * A143924 A143925 A143926

Keyword

nonn

Author

Paul D. Hanna, Sep 06 2008

Status

approved