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A058014
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Number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.
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7
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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; internal format)
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OFFSET
| 0,4
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REFERENCES
| A. Postnikov and R. P. Stanley: Deformations of Coxeter hyperplane arrangements, J. Combin. Theory, Ser. A, 91 (2000), 544-597. (Section 10.2.)
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LINKS
| Alexander Postnikov, Papers.
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FORMULA
| 2^{-n} Sum_{k=0}^n {n! \over k! (n-k)!} (n+1-2k)^{n-1}
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), 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)). [From Vladimir Kruchinin (kru(AT)ie.tusur.ru), Apr 19 2011]
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EXAMPLE
| E.g.f. A(x) = 1 + x + x^2/2! + 4x^3/3! + 13x^4/4! + 96x^5/5! +...
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MAPLE
| b := (n)->2^(-n)*sum('binomial(n, k)*(n+1-2*k)^(n-1)', 'k'=0..n);
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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}] (* From Jean-François Alcover, Nov 14 2011, after Maple *)
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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 (pauldhanna(AT)juno.com), Mar 29 2008
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CROSSREFS
| Cf. bisections: A007106, A143601.
Cf. A138764 (variant).
Sequence in context: A131590 A200850 A041433 * A045886 A015460 A121813
Adjacent sequences: A058011 A058012 A058013 * A058015 A058016 A058017
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KEYWORD
| easy,nice,nonn
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AUTHOR
| Alex Postnikov (apost(AT)math.mit.edu), Nov 13 2000
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EXTENSIONS
| Updated URL and author's e-mail address - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 23 2010
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