%I #33 Jan 30 2020 21:29:17

%S 0,1,3,14,70,363,1925,10364,56412,309605,1710247,9496746,52960674,

%T 296408847,1663998345,9365980152,52837614456,298676661129,

%U 1691325089867,9592607927750,54482777049918,309837754937843,1764046900535053,10054065679046004,57357471874390100

%N Expansion of (x-1)/8 - (x^2-4*x-1)/(8*sqrt(x^2-6*x+1)).

%C Number of vertices in all Schroeder trees with n leaves. See Theorem 2.1 of Van Duzer. - _Michel Marcus_, Apr 12 2019

%H G. C. Greubel, <a href="/A261207/b261207.txt">Table of n, a(n) for n = 0..1000</a>

%H Anthony Van Duzer, <a href="https://arxiv.org/abs/1904.05525">Subtrees of a Given size in Schroeder Trees</a>, arXiv:1904.05525 [math.CO], 2019.

%F a(n) = Sum_{i=0..n-1}(2^i*(-1)^(n-i-1)*C(n+1,n-i-1)*C(n+i,n)).

%F a(n) = (-1)^(n+1)*(n*(n+1)/2)*hypergeom([1-n, 1+n], [3], 2)). - _Peter Luschny_, Aug 12 2015

%F a(n) = A010683(n-1)*(n+1)/2. - _Peter Luschny_, Aug 12 2015

%F a(n) ~ (3+2*sqrt(2))^n / (2^(9/4)*sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 17 2015

%F D-finite with recurrence: n*a(n) +(-2*n-5)*a(n-1) +3*(-8*n+21)*a(n-2) +(10*n-39)*a(n-3) +(-n+5)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2020

%p a := n -> simplify((-1)^(n+1)*(n*(n+1)/2)*hypergeom([1-n, 1+n], [3], 2));

%p seq(a(n),n=0..27); # _Peter Luschny_, Aug 12 2015

%t CoefficientList[Series[(x - 1) / 8 - (x^2 - 4 x - 1) / (8 Sqrt[x^2 - 6 x + 1]), {x, 0, 33}], x] (* _Vincenzo Librandi_, Aug 12 2015 *)

%o (Maxima) a(n):=sum(2^i*(-1)^(n-i-1)*binomial(n+1,n-i-1)*binomial(n+i,n),i,0,n-1);

%o (PARI) vector(30, n, n--; sum(i=0,n-1,2^i*(-1)^(n-i-1)*binomial(n+1,n-i-1)*binomial(n+i,n))) \\ _Michel Marcus_, Aug 12 2015

%Y Cf. A010683.

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Aug 11 2015