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A007106
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Number of labeled odd degree trees with 2n nodes.
(Formerly M3704)
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15
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1, 4, 96, 5888, 686080, 130179072, 36590059520, 14290429935616, 7405376630685696, 4917457306800619520, 4071967909087792857088, 4113850542422629363482624, 4980673081258443273955966976, 7119048451600750435732824260608, 11861520124846917915630931846103040
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OFFSET
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1,2
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REFERENCES
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R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Bisection of A058014. The expansion 1/sqrt(1+x^2)*arcsinh(x) = x - 4*x^3/3! + 64*x^5/5! - ... (see A002454) has series reversion x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + .... The coefficients appear to be the terms of this sequence. As an x-adic limit this e.g.f. equals lim_{n -> infinity} sinh(f(n,x)), where f(0,x) = x and f(n,x) = x*cosh(f(n-1,x)) for n >= 1. See the example section below. - Peter Bala, Apr 24 2012
a(n) = Sum_{k=1..n} binomial(n,k) * k! * (n-2)! [z^{n-2}] [u^k] exp(u(exp(z)+exp(-z)-2)/2)). - Marko Riedel, Jun 16 2016
a(n) = (1/2) * Sum_{k=0..n-1} binomial(2*n,k) * (n-k)^(2*n-2) for n >= 2.
a(n) = (2*n-1)!*[x^(2*n-1)] sinh(REVERT(x/cosh(x))), see A036778. (End)
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EXAMPLE
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Let G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + ... be the e.g.f. for A143601. Then sinh(x*G(x)) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + ....
Conjectural e.g.f. as an x-adic limit:
sinh(x) = x + ...; sinh(x*cosh(x)) = x + 4*x^3/3! + ...;
sinh(x*cosh(x*cosh(x))) = x + 4*x^3/3! + 96*x^5/5! + ...;
sinh(x*cosh(x*cosh(x*cosh(x)))) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + ....
(End)
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MAPLE
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A007106(n) = A(2n) where n>=2, A(n) = (add(binomial(n, q)*(n-2*q)^(n-2)/(n-2)!, q=0..n) - add(binomial(n-1, q)*(n-2*q)^(n-3)/(n-3)!, q=0..n-1) + add(binomial(n-1, q)*(n-2-2*q)^(n-3)/(n-3)!, q=0..n-1))*n!/2^(n+1)/(n-1)
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MATHEMATICA
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{1}~Join~Array[(1/2)*Sum[Binomial[2 #, k]*(# - k)^(2 # - 2), {k, 0, # - 1}] &, 12, 2] (* Michael De Vlieger, Oct 13 2021 *)
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PROG
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(PARI) a(n) = if(n<=1, n==1, sum(k=0, n-1, binomial(2*n, k) * (n-k)^(2*n-2))/2) \\ Andrew Howroyd, Nov 22 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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