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

%I A060279 #28 Nov 06 2024 04:14:36
%S A060279 2,16,576,47104,6860800,1562148864,512260833280,228646878969856,
%T A060279 133296779352342528,98349146136012390400,89583293999931442855936,
%U A060279 98732413018143104723582976,129497500112719525122855141376,199333356644821012200519079297024
%N A060279 Number of labeled rooted trees with all 2n nodes of odd degree.
%C A060279 There are no such trees with an odd number of nodes.
%D A060279 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
%H A060279 G. C. Greubel, <a href="/A060279/b060279.txt">Table of n, a(n) for n = 1..210</a>
%F A060279 a(n) = (n/2^n)*Sum_{k=0..n} binomial(n, k)*(n-2*k)^(n-2).
%F A060279 a(n) = 2*n * A007106(n).
%F A060279 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
%p A060279 a:= j-> (n-> (n/2^n)*add(binomial(n, k)*(n-2*k)^(n-2), k=0..n))(2*j):
%p A060279 seq(a(n), n=1..15);  # _Alois P. Heinz_, Sep 27 2020
%t A060279 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 *)
%t A060279 A060279[n_]:= n*Sum[Binomial[2*n,k]*(n-k)^(2*n-2), {k,0,n-1}] +Boole[n==1];
%t A060279 Table[A060279[n], {n,40}] (* _G. C. Greubel_, Nov 05 2024 *)
%o A060279 (PARI) a(n) = n/2^n*sum(k=0, n, binomial(n, k)*(n-2*k)^(n-2)) \\ _Michel Marcus_, Jun 17 2013
%o A060279 (Magma)
%o A060279 A060279:= func< n | n eq 1 select 2 else n*(&+[Binomial(2*n,k)*(n-k)^(2*n-2) : k in [0..n-1]]) >;
%o A060279 [A060279(n): n in [1..30]]; // _G. C. Greubel_, Nov 05 2024
%o A060279 (SageMath)
%o A060279 def A060279(n): return n*sum( binomial(2*n,k)*(n-k)^(2*n-2) for k in range(n)) + int(n==1)
%o A060279 [A060279(n) for n in range(1,41)] # _G. C. Greubel_, Nov 05 2024
%Y A060279 Cf. A007106.
%K A060279 easy,nonn
%O A060279 1,1
%A A060279 _Vladeta Jovovic_, Mar 28 2001