OFFSET
1,1
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
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
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 *)
PROG
(PARI) a(n) = n/2^n*sum(k=0, n, binomial(n, k)*(n-2*k)^(n-2)) \\ Michel Marcus, Jun 17 2013
(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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Mar 28 2001
STATUS
approved