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A200318 E.g.f. satisfies: A(x) = x-1 + cosh(A(x)).

%I A200318 #29 Aug 01 2016 09:52:47
%S A200318 1,1,3,16,120,1156,13608,189316,3039060,55291336,1124309208,
%T A200318 25268818576,622008616320,16642670404816,480923246983728,
%U A200318 14926731083999296,495243684302520000,17491488288340789696,655224017429959987968,25947019896579324410176,1083050878686674070676800
%N A200318 E.g.f. satisfies: A(x) = x-1 + cosh(A(x)).
%C A200318 a(n) is the number of leaf labeled rooted trees with n leaves in which the outdegrees of the root and all internal nodes are positive even integers. - _Geoffrey Critzer_, Jul 31 2016
%H A200318 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 128,(labeled hierarchies).
%F A200318 E.g.f. satisfies:
%F A200318 (1) A(x) = Series_Reversion(1+x - cosh(x)).
%F A200318 (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (cosh(x) - 1)^n / n!.
%F A200318 (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (cosh(x) - 1)^n/x / n! ).
%F A200318 a(n) ~ n^(n-1) / (2^(1/4) * exp(n) * (1-sqrt(2)+log(1+sqrt(2)))^(n-1/2)). - _Vaclav Kotesovec_, Jan 10 2014
%e A200318 E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 16*x^4/4! + 120*x^5/5! +...
%e A200318 where A(1+x - cosh(x)) = x and A(x) = x-1 + cosh(A(x)).
%e A200318 The e.g.f. satisfies:
%e A200318 A(x) = x + (cosh(x)-1) + d/dx (cosh(x)-1)^2/2! + d^2/dx^2 (cosh(x)-1)^3/3! + d^3/dx^3 (cosh(x)-1)^4/4! +...
%e A200318 as well as the logarithmic series:
%e A200318 log(A(x)/x) = (cosh(x)-1)/x + d/dx (cosh(x)-1)^2/x/2! - d^2/dx^2 (cosh(x)-1)^3/x/3! + d^3/dx^3 (cosh(x)-1)^4/x/4! +...
%t A200318 Rest[CoefficientList[InverseSeries[Series[1 + x - Cosh[x],{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 10 2014 *)
%o A200318 (PARI) {a(n)=n!*polcoeff(serreverse(1+x-cosh(x+x^2*O(x^n))),n)}
%o A200318 for(n=1, 21, print1(a(n), ", "))
%o A200318 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A200318 {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, (cosh(x+x*O(x^n))-1)^m)/m!)+x*O(x^n)); n!*polcoeff(A, n)}
%o A200318 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A200318 {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, (cosh(x+x*O(x^n))-1)^m/x)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
%Y A200318 Cf. A200317, A000311, A052526.
%K A200318 nonn
%O A200318 1,3
%A A200318 _Paul D. Hanna_, Nov 15 2011