A007106 Number of labeled odd degree trees with 2n nodes. (Formerly M3704)

1, 4, 96, 5888, 686080, 130179072, 36590059520, 14290429935616, 7405376630685696, 4917457306800619520, 4071967909087792857088, 4113850542422629363482624, 4980673081258443273955966976, 7119048451600750435732824260608, 11861520124846917915630931846103040

Offset

1,2

References

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).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..211 (terms 1..39 from R. W. Robinson)

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.

Mathematics Stack Exchange, Marko R. Riedel, Odd degree trees

Mathematics Stack Exchange, Marko R. Riedel, Odd degree trees II

Marko Riedel, Count by Prüfer codes and Stirling numbers

Index entries for sequences related to trees

Formula

a(n) = A060279(n)/(2*n). - Vladeta Jovovic, Feb 08 2005

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

From Alexander Burstein, Oct 13 2021: (Start)

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)

a(n) = Sum_{k=0..n-1} A156289(n-1, k)*(2*n)!/(2*n-k)!. - Peter Luschny, May 07 2022

Example

From Peter Bala, Apr 24 2012: (Start)
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)

Maple

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)

Mathematica

{1}~Join~Array[(1/2)*Sum[Binomial[2 #, k]*(# - k)^(2 # - 2), {k, 0, # - 1}] &, 12, 2] (* Michael De Vlieger, Oct 13 2021 *)

Prog

(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

Crossrefs

Cf. A036778, A058014, A060279, A143601, A156289.

Sequence in context: A013042 A190196 A065140 * A111637 A027872 A308146

Adjacent sequences: A007103 A007104 A007105 * A007107 A007108 A007109

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Corrected and extended by Vladeta Jovovic, Feb 08 2005

Status

approved