|
%I #8 Oct 07 2020 03:32:56
%S 1,1,4,27,264,3400,54480,1045800,23412480,599157216,17258814720,
%T 552733695360,19485393903360,749871707270400,31283408387911680,
%U 1406370859616923200,67780975948945459200,3486485719168394342400,190644828634476331315200,11043310871932837194977280
%N E.g.f. satisfies: A(x) = 1 + log(1 + x^2*A(x)^2)/x.
%H Vaclav Kotesovec, <a href="/A218653/b218653.txt">Table of n, a(n) for n = 0..300</a>
%F E.g.f. satisfies: A(x - log(1+x^2)) = x/(x - log(1+x^2)).
%F E.g.f.: A(x) = (1/x)*Series_Reversion(x - log(1+x^2)).
%F a(n) = A218652(n+1)/(n+1).
%F a(n) ~ Gamma(1/3) * n^(n - 5/6) / (6^(1/6) * sqrt(Pi) * exp(n) * (1 - log(2))^(n + 2/3)). - _Vaclav Kotesovec_, Oct 07 2020
%e E.g.f: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 264*x^4/4! + 3400*x^5/5! +...
%e Related expansions:
%e A(x)^2 = 1 + 2*x + 10*x^2/2! + 78*x^3/3! + 840*x^4/4! + 11600*x^5/5! +...
%e log(1 + x^2*A(x)^2)/x = x + 4*x^2/2! + 27*x^3/3! + 264*x^4/4! + 3400*x^5/5! +...
%o (PARI) {a(n)=n!*polcoeff((1/x)*serreverse(x-log(1+x^2 +x^2*O(x^n))), n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A218652, A213641.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 03 2012
| 