OFFSET
0,2
COMMENTS
Central coefficients T(2n,n) of the Riordan array ((1-x)/(1-2x), x(1-x)/(1-2x)), A105306.
a(n) counts the bi-degree sequences of directed trees (i.e., digraphs whose underlying graph is a tree) with n edges. - Nikos Apostolakis, Dec 31 2016
a(n) is also the number of Dyck paths having exactly n peaks in level 1 and n peaks in level 2 and no other peaks. a(2) = 9: /\/\//\/\\, /\//\/\\/\, //\/\\/\/\, /\/\//\\//\\, /\//\\/\//\\, /\//\\//\\/\, //\\/\/\//\\, //\\/\//\\/\, //\\//\\/\/\. - Alois P. Heinz, Jun 20 2017
For n>0, a(n) is the number of ordered trees with n+1 leaves, no node of outdegree 1, and having one of its leaves marked. - Juan B. Gil, Jan 03 2024
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021).
Juan B. Gil, Emma G. Hoover, and Jessica A. Shearer, Bijections between colored compositions, Dyck paths, and polygon partitions, arXiv:2403.04575 [math.CO], 2024.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
V. V. Kruchinin and D. V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, arXiv preprint arXiv:1206.0877 [math.CO], 2012, and J. Int. Seq. 15 (2012) #12.9.3
FORMULA
E.g.f.: 1+exp(3*x)*Bessel_I(1,2*sqrt(2)*x)/sqrt(2) +int(exp(3*x) *Bessel_I(1,2*sqrt(2)*x) /(sqrt(2)*x),x).
G.f.: 1/4 - (x-3)/(4*sqrt(x^2-6*x+1)). - Dmitry Kruchinin, Aug 31 2012
Conjecture: n*(n-1)*a(n) -3*(2*n-1)*(n-1)*a(n-1) +n*(n-2)*a(n-2) = 0. - R. J. Mathar, Dec 03 2014
a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(n+k,n). - Nikos Apostolakis, Dec 31 2016
a(n) = (n+1)*hypergeom([1-n, -n], [2], 2). - Peter Luschny, Jan 02 2017
MAPLE
a:= proc(n) option remember; `if`(n<2, n+1,
(6*n-3)/n*a(n-1) -(n-2)/(n-1)*a(n-2))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Jun 22 2017
MATHEMATICA
a[n_] := Sum[Binomial[n - 1, k - 1]*Binomial[n + k, n], {k, 0, n}]; Array[a, 25, 0] (* or *)
CoefficientList[ Series[1/4 - (x - 3)/(4 Sqrt[x^2 - 6x +1]), {x, 0, 25}], x] (* Robert G. Wilson v, Dec 31 2016 *)
Table[(n+1)Hypergeometric2F1[1-n, -n, 2, 2], {n, 0, 21}] (* Peter Luschny, Jan 02 2017 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 18 2010
STATUS
approved