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%I #19 Sep 08 2022 08:45:56
%S 0,2,3,14,45,167,609,2270,8517,32207,122463,467843,1794195,6903353,
%T 26635773,103020254,399300165,1550554583,6031074183,23493410759,
%U 91638191235,357874310213,1399137067683,5475504511859,21447950506395,84083979575117
%N Binomial sum related to rooted trees.
%H G. C. Greubel, <a href="/A188289/b188289.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = binomial(2*n,n) - (-1)^n - Sum_{k=0..n-1} binomial(2*k, n-1).
%F a(n) = Sum_{k=1..n} binomial(n+k,k)*(Sum_{r=n-k..n} (-1)^r binomial(n-k, r).
%F a(n) = (-1)^n*2^(-(1+n))*(1 - 2^(1+n) + (-2)^n*binomial(2+2*n, 1+n) * hypergeometric2F1(1, 2+2*n; 2+n; -1).
%t Table[Binomial[2n,n]-(-1)^n-Sum[Binomial[2k,n-1],{k,0,n-1}],{n,0,30}] (* _Harvey P. Dale_, Dec 10 2012 *)
%o (PARI) {a(n) = binomial(2*n,n) -(-1)^n -sum(k=0,n-1, binomial(2*k,n-1))}; \\ _G. C. Greubel_, Apr 29 2019
%o (Magma) [n eq 0 select 0 else Binomial(2*n, n) -(-1)^n - (&+[Binomial(2*k, n-1): k in [0..n-1]]): n in [0..30]]; // _G. C. Greubel_, Apr 29 2019
%o (Sage) [binomial(2*n,n) -(-1)^n -sum(binomial(2*k, n-1) for k in (0..n-1)) for n in (0..30)] # _G. C. Greubel_, Apr 29 2019
%o (GAP) List([0..30], n-> Binomial(2*n,n) -(-1)^n -Sum([0..n-1], k-> Binomial(2*k,n-1))) # _G. C. Greubel_, Apr 29 2019
%Y Cf. A000984, A014300, A026641, A178792, A176479, A072547.
%K nonn
%O 0,2
%A _Olivier Gérard_, Aug 19 2012
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