A144009 E.g.f. satisfies: A'(x) = (1 + x*A(x))^4 with A(0)=1.

1, 1, 4, 20, 144, 1352, 15360, 206688, 3214848, 56694144, 1118486016, 24409113600, 583825803264, 15188350556160, 426989455147008, 12899931159564288, 416802018563850240, 14342136885537472512, 523630043964811247616

Offset

0,3

Comments

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

if F(x) = 1/(1-x) then F'(x) = (1 + x*F(x))^2.

In general, if e.g.f satisfies A'(x) = (1+x*A(x))^p, then a(n) ~ c(p) * d(p)^n * n! / n^(1-1/(p-1)), where c(p) and d(p) are constants independent on n. - Vaclav Kotesovec, Jul 15 2014

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Formula

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

a(n) ~ c * n^(n-1/6) / (exp(n) * r^n), where r = 0.475460695778... and c = 2.2399022393... . - Vaclav Kotesovec, Jul 14 2014

Mathematica

n = 18; A = 1+x; Do[A = 1 + Integrate[(1+x*A)^4 + O[x]^n, x], {i, 0, n}]; CoefficientList[A, x]*Range[0, n]! (* Jean-François Alcover, Jul 20 2017, adapted from PARI *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+intformal((1+x*A+x*O(x^n))^4)); n!*polcoeff(A, n)}

Crossrefs

Cf. A144008, A245247, A245248, A245249.

Sequence in context: A004204 A160567 A034216 * A301270 A366183 A117887

Adjacent sequences: A144006 A144007 A144008 * A144010 A144011 A144012

Keyword

nonn

Author

Paul D. Hanna, Sep 10 2008

Status

approved