A261207 Expansion of (x-1)/8 - (x^2-4*x-1)/(8*sqrt(x^2-6*x+1)).

0, 1, 3, 14, 70, 363, 1925, 10364, 56412, 309605, 1710247, 9496746, 52960674, 296408847, 1663998345, 9365980152, 52837614456, 298676661129, 1691325089867, 9592607927750, 54482777049918, 309837754937843, 1764046900535053, 10054065679046004, 57357471874390100

Offset: 0

Author

Vladimir Kruchinin, Aug 11 2015

Keywords

• nonn

Comments

Number of vertices in all Schroeder trees with n leaves. See Theorem 2.1 of Van Duzer. - Michel Marcus, Apr 12 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Anthony Van Duzer, Subtrees of a Given size in Schroeder Trees, arXiv:1904.05525 [math.CO], 2019.

Crossrefs

Cf. A010683.

Programs

Maple

a := n -> simplify((-1)^(n+1)*(n*(n+1)/2)*hypergeom([1-n, 1+n], [3], 2));
seq(a(n),n=0..27); # Peter Luschny, Aug 12 2015

Mathematica

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 *)

Maxima

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

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

Formula

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

a(n) = (-1)^(n+1)*(n*(n+1)/2)*hypergeom([1-n, 1+n], [3], 2). - Peter Luschny, Aug 12 2015

a(n) = A010683(n-1)*(n+1)/2. - Peter Luschny, Aug 12 2015

a(n) ~ (3+2*sqrt(2))^n / (2^(9/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 17 2015

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