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A144009 E.g.f. satisfies: A'(x) = (1 + x*A(x))^4 with A(0)=1. 5

%I #14 Jul 20 2017 10:42:31

%S 1,1,4,20,144,1352,15360,206688,3214848,56694144,1118486016,

%T 24409113600,583825803264,15188350556160,426989455147008,

%U 12899931159564288,416802018563850240,14342136885537472512,523630043964811247616

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

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

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

%C 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

%H Vaclav Kotesovec, <a href="/A144009/b144009.txt">Table of n, a(n) for n = 0..250</a>

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

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

%t 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 *)

%o (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)}

%Y Cf. A144008, A245247, A245248, A245249.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 10 2008

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Last modified March 29 08:00 EDT 2024. Contains 371265 sequences. (Running on oeis4.)