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A058014 Number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.

%I A058014 #51 Jul 05 2025 09:56:25
%S A058014 1,1,1,4,13,96,541,5888,47545,686080,7231801,130179072,1695106117,
%T A058014 36590059520,567547087381,14290429935616,257320926233329,
%U A058014 7405376630685696,151856004814953841,4917457306800619520,113144789723082206461,4071967909087792857088
%N A058014 Number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.
%H A058014 Seiichi Manyama, <a href="/A058014/b058014.txt">Table of n, a(n) for n = 0..422</a>
%H A058014 Alexander Postnikov, <a href="http://math.mit.edu/~apost/papers.html">Papers</a>.
%H A058014 A. Postnikov and R. P. Stanley, <a href="http://dx.doi.org/10.1006/jcta.2000.3106">Deformations of Coxeter hyperplane arrangements</a>, J. Combin. Theory, Ser. A, 91 (2000), 544-597. (Section 10.2.)
%F A058014 a(n) = (1/2^n) * Sum_{k=0..n} binomial(n,k) * (n + 1 - 2*k)^(n-1).
%F A058014 From _Paul D. Hanna_, Mar 29 2008: (Start)
%F A058014 E.g.f. satisfies A(x) = exp( x*[A(x) + 1/A(x)]/2 ).
%F A058014 E.g.f. A(x) equals the inverse function of 2*x*log(x)/(1 + x^2).
%F A058014 Let r = radius of convergence of A(x), then r = 0.6627434193491815809747420971092529070562335491150224... and A(r) = 3.31905014223729720342271370055697247448941708369151595... where A(r) and r satisfy A(r) = exp( (A(r)^2 + 1)/(A(r)^2 - 1) ) and r = 2*A(r)/(A(r)^2 - 1). (End)
%F A058014 E.g.f. A(x)=exp(B(x)), B(x) satisfies B(x)=x*cosh(B(x)). [_Vladimir Kruchinin_, Apr 19 2011]
%F A058014 a(n) ~ (1-(-1)^n*s^2)/s * n^(n-1) * ((1-s^2)/(2*s))^n / exp(n), where s = 0.3012910191606573456... is the root of the equation (1+s^2) = (s^2-1)*log(s), r = 2*s/(1-s^2). - _Vaclav Kotesovec_, Jan 08 2014
%F A058014 E.g.f. satisfies A(-x) = 1/A(x). - _Alexander Burstein_, Oct 26 2021
%F A058014 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (2*k+1) * binomial(n-1,2*k) * a(2*k) * a(n-1-2*k). - _Seiichi Manyama_, Jul 05 2025
%e A058014 E.g.f.: A(x) = 1 + x + x^2/2! + 4x^3/3! + 13x^4/4! + 96x^5/5! +...
%p A058014 a := n -> 2^(-n)*add(binomial(n,k)*(n+1-2*k)^(n-1), k=0..n);
%t A058014 a[n_] := Sum[((n-2k+1)^(n-1)*n!) / (k!*(n-k)!), {k, 0, n}] / 2^n; a[1] = 1; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Nov 14 2011, after Maple *)
%o A058014 (PARI) {a(n)=local(A=1+x);for(i=0,n,A=exp(x*(A+1/(A +x*O(x^n)))/2));n!*polcoeff(A,n)} \\ _Paul D. Hanna_, Mar 29 2008
%o A058014 (PARI) {a(n) = sum(k=0, n, binomial(n, k)*(n+1-2*k)^(n-1))/2^n} \\ _Seiichi Manyama_, Sep 27 2020
%Y A058014 Cf. bisections: A007106, A143601.
%Y A058014 Cf. A138764 (variant).
%K A058014 easy,nice,nonn
%O A058014 0,4
%A A058014 Alex Postnikov (apost(AT)math.mit.edu), Nov 13 2000
%E A058014 Updated URL and author's e-mail address - _R. J. Mathar_, May 23 2010