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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. 9
 1, 1, 1, 4, 13, 96, 541, 5888, 47545, 686080, 7231801, 130179072, 1695106117, 36590059520, 567547087381, 14290429935616, 257320926233329, 7405376630685696, 151856004814953841, 4917457306800619520 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Alexander Postnikov, Papers. A. Postnikov and R. P. Stanley, Deformations of Coxeter hyperplane arrangements, J. Combin. Theory, Ser. A, 91 (2000), 544-597. (Section 10.2.) FORMULA a(n) = 2^(-n) * sum(k=0..n, n!/(k!*(n-k)!) * (n + 1 - 2*k)^(n-1) ). From Paul D. Hanna, Mar 29 2008: (Start) E.g.f. satisfies A(x) = exp( x*[A(x) + 1/A(x)]/2 ). E.g.f. A(x) equals the inverse function of 2*x*log(x)/(1 + x^2). 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) E.g.f. A(x)=exp(B(x)), B(x) satisfies B(x)=x*cosh(B(x)). [Vladimir Kruchinin, Apr 19 2011] 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 EXAMPLE E.g.f.: A(x) = 1 + x + x^2/2! + 4x^3/3! + 13x^4/4! + 96x^5/5! +... MAPLE b := (n)->2^(-n)*sum('binomial(n, k)*(n+1-2*k)^(n-1)', 'k'=0..n); MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. bisections: A007106, A143601. Cf. A138764 (variant). Sequence in context: A200850 A041433 A222764 * A290392 A261785 A276912 Adjacent sequences:  A058011 A058012 A058013 * A058015 A058016 A058017 KEYWORD easy,nice,nonn AUTHOR Alex Postnikov (apost(AT)math.mit.edu), Nov 13 2000 EXTENSIONS Updated URL and author's e-mail address - R. J. Mathar, May 23 2010 STATUS approved

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Last modified June 17 19:10 EDT 2019. Contains 324198 sequences. (Running on oeis4.)