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E.g.f. satisfies: A(x) = exp(Integral(1+x*A(x)^4) dx), where the constant of integration is zero.
4

%I #23 Sep 23 2022 17:00:20

%S 1,1,2,12,102,1062,13812,215592,3896892,80103612,1847079192,

%T 47204854992,1324132604232,40446893218632,1336423937927472,

%U 47492006442366432,1806200688076918032,73199329659111178512,3149155288463030836512,143338650123433404564672

%N E.g.f. satisfies: A(x) = exp(Integral(1+x*A(x)^4) dx), where the constant of integration is zero.

%C In general, if e.g.f. satisfies: A(x) = exp( Integral(1 + x*A(x)^p) dx ), p>1, and the constant of integration is zero, then A(x) = (1/p + (p-1)/(exp(p*x)*p) - x)^(-1/p), and a(n) ~ n! * p^(n+1/p) / (Gamma(1/p) * n^(1-1/p)* (1+LambertW((p-1)*exp(-1)))^(n+2/p)).

%H G. C. Greubel, <a href="/A245266/b245266.txt">Table of n, a(n) for n = 0..385</a>

%F E.g.f.: 4^(1/4)*exp(x)/(exp(4*x) - 4*exp(4*x)*x + 3)^(1/4).

%F a(n) ~ Gamma(3/4) * 2^(2*n+1/2) * n^(n-1/4) / (sqrt(Pi) * exp(n) * (1+LambertW(3/exp(1)))^(n+1/2)). - _Vaclav Kotesovec_, Jul 15 2014

%t CoefficientList[Series[(1/p + (p-1)/(E^(p*x)*p) - x)^(-1/p) /. p->4, {x, 0, 20}], x] * Range[0, 20]!

%o (PARI) x='x+O('x^30); round(Vec(serlaplace(4^(1/4)*exp(x)/(exp(4*x) - 4*exp(4*x)*x + 3)^(1/4)))) \\ _G. C. Greubel_, Nov 21 2017

%Y Cf. A212913 (p=2), A212914 (p=3), A245267 (p=5).

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Jul 15 2014