OFFSET
0,3
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.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Aug 11 2015
STATUS
approved