Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #36 Nov 16 2024 02:02:35
%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