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%I #26 Mar 04 2024 11:52:13
%S 1,1,13,541,47545,7231801,1695106117,567547087381,257320926233329,
%T 151856004814953841,113144789723082206461,103890621918675777804301,
%U 115270544419577901796226473,152049571406030636219959644841
%N Number of labeled odd-degree trees with 2n+1 nodes.
%H Seiichi Manyama, <a href="/A143601/b143601.txt">Table of n, a(n) for n = 0..211</a>
%F E.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! satisfies the following formulas.
%F (1) A(x) = cosh(x*A(x)).
%F (2) A(x) = (1/x)*Series_Reversion( x/cosh(x) ).
%F (3) sqrt(A(x)^2 - 1) = e.g.f. of A007106.
%F (4) exp(x*A(x)) = A(x) + sqrt(A(x)^2-1) = e.g.f. of A058014.
%F (5) A(x) = [F(x) + F(-x)]/2 where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
%F (6) A(2*x) = [G(x)/G(-x) + G(-x)/G(x)]/2 where G(x) = exp(x*G(x)/G(-x)) = e.g.f. of A143600.
%F From _Paul D. Hanna_, Aug 29 2008: (Start)
%F (7) A(x/cosh(x)) = cosh(x).
%F (8) a(n) = (2n)!*[x^(2n)] cosh(x)^(2n+1)/(2n+1). (End)
%F (9) a(n) = Sum_{k=0..n} binomial(2*n+1,k) * (2*n+1 - 2*k)^(2*n) / ((2*n+1) * 2^(2*n)). [See formula by _Christophe Vignat_ in A309204.] - _Paul D. Hanna_, Feb 19 2024
%F a(n) ~ 2^(2*n) * n^(2*n-1) * (s^2-1)^(n+1/2) / exp(2*n), where s = 1.810170580698977274512829... is the root of the equation sqrt(s^2-1) * log(s + sqrt(s^2-1)) = s. - _Vaclav Kotesovec_, Jan 10 2014
%F Radius of convergence r = 0.66274341934918158097474... = 1/sqrt(s^2-1) and A(r) = s (given above) satisfy r = 1/sinh(r*A(r)) and A(r) = cosh(r*A(r)). - _Paul D. Hanna_, Mar 04 2024
%e E.g.f.: A(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! + ...
%e The e.g.f. of A007106 (a bisection of A058014) is given by:
%e sqrt(A(x)^2 - 1) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! + ...
%e The e.g.f. of A058014 is given by:
%e F(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! + ...
%e where A(x) = [F(x) + F(-x)]/2 and exp(x*A(x)) = F(x).
%e The e.g.f. of A143600 is given by:
%e G(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! + ...
%e where A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2.
%t Table[(2*n)!*CoefficientList[1/x*InverseSeries[Series[x/Cosh[x],{x,0,41}],x],x][[2*n+1]],{n,0,20}] (* _Vaclav Kotesovec_, Jan 10 2014 *)
%o (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=cosh(x*A));n!*polcoeff(A,n)}
%o (PARI) {a(n)=(2*n)!*polcoeff(cosh(x+x*O(x^(2*n)))^(2*n+1)/(2*n+1),2*n)} \\ _Paul D. Hanna_, Aug 29 2008
%o (PARI) {a(n) = sum(k=0,n, binomial(2*n+1,k) * (2*n+1-2*k)^(2*n) / ((2*n+1) * 2^(2*n)) )}
%o for(n=0,30, print1(a(n),", ")) \\ _Paul D. Hanna_, Feb 19 2024
%Y Cf. A058014, A143600, A007106, A309204.
%Y Cf. A370430, A370432.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 26 2008, May 27 2009
%E Edited by _Paul D. Hanna_, May 27 2009
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