A060279 Number of labeled rooted trees with all 2n nodes of odd degree.

2, 16, 576, 47104, 6860800, 1562148864, 512260833280, 228646878969856, 133296779352342528, 98349146136012390400, 89583293999931442855936, 98732413018143104723582976, 129497500112719525122855141376, 199333356644821012200519079297024

Offset: 1

Author

Vladeta Jovovic, Mar 28 2001

Keywords

• easy
• nonn

Comments

There are no such trees with an odd number of nodes.

References

• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..210

Crossrefs

Cf. A007106.

Programs

Magma

A060279:= func< n | n eq 1 select 2 else n*(&+[Binomial(2*n,k)*(n-k)^(2*n-2) : k in [0..n-1]]) >;
[A060279(n): n in [1..30]]; // G. C. Greubel, Nov 05 2024

Maple

a:= j-> (n-> (n/2^n)*add(binomial(n, k)*(n-2*k)^(n-2), k=0..n))(2*j):
seq(a(n), n=1..15);  # Alois P. Heinz, Sep 27 2020

Mathematica

Flatten[{2,Table[n/2^n*Sum[Binomial[n,k]*(n-2*k)^(n-2),{k,0,n}],{n,4,30,2}]}] (* Vaclav Kotesovec, Jan 23 2014 *)
A060279[n_]:= n*Sum[Binomial[2*n,k]*(n-k)^(2*n-2), {k,0,n-1}] +Boole[n==1];
Table[A060279[n], {n,40}] (* G. C. Greubel, Nov 05 2024 *)

PARI

a(n) = n/2^n*sum(k=0, n, binomial(n, k)*(n-2*k)^(n-2)) \\ Michel Marcus, Jun 17 2013

SageMath

def A060279(n): return n*sum( binomial(2*n,k)*(n-k)^(2*n-2) for k in range(n)) + int(n==1)
[A060279(n) for n in range(1,41)] # G. C. Greubel, Nov 05 2024

Formula

a(n) = (n/2^n)*Sum_{k=0..n} binomial(n, k)*(n-2*k)^(n-2).

a(n) = 2*n * A007106(n).

a(n) ~ sqrt(1+s^2) * s^(2*n-1) * 2^(2*n) * n^(2*n-1) / exp(2*n), where s = 1.5088795615383199289... is the root of the equation sqrt(1+s^2) = s*log(s+sqrt(1+s^2)). - Vaclav Kotesovec, Jan 23 2014