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A176479 a(n) = (n+1)*A001003(n).

%I A176479 #72 Mar 08 2024 01:17:36
%S A176479 1,2,9,44,225,1182,6321,34232,187137,1030490,5707449,31760676,
%T A176479 177435297,994551222,5590402785,31500824304,177880832001,
%U A176479 1006362234162,5703029112297,32367243171740,183945502869345,1046646207221582,5961966567317649,33995080211156904
%N A176479 a(n) = (n+1)*A001003(n).
%C A176479 Central coefficients T(2n,n) of the Riordan array ((1-x)/(1-2x), x(1-x)/(1-2x)), A105306.
%C A176479 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
%C A176479 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
%C A176479 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
%H A176479 Alois P. Heinz, <a href="/A176479/b176479.txt">Table of n, a(n) for n = 0..1000</a>
%H A176479 Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, <a href="https://doi.org/10.26493/1855-3974.2478.d1b">Enumerating symmetric peaks in non-decreasing Dyck paths</a>, Ars Mathematica Contemporanea (2021).
%H A176479 Juan B. Gil, Emma G. Hoover, and Jessica A. Shearer, <a href="https://arxiv.org/abs/2403.04575">Bijections between colored compositions, Dyck paths, and polygon partitions</a>, arXiv:2403.04575 [math.CO], 2024.
%H A176479 Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H A176479 V. V. Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1206.0877">A Method for Obtaining Generating Function for Central Coefficients of Triangles</a>, arXiv preprint arXiv:1206.0877 [math.CO], 2012, and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kruchinin/kruchinin5.html">J. Int. Seq. 15 (2012) #12.9.3</a>
%F A176479 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).
%F A176479 G.f.: 1/4 - (x-3)/(4*sqrt(x^2-6*x+1)). - _Dmitry Kruchinin_, Aug 31 2012
%F A176479 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
%F A176479 a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(n+k,n). - _Nikos Apostolakis_, Dec 31 2016
%F A176479 a(n) = (n+1)*hypergeom([1-n, -n], [2], 2). - _Peter Luschny_, Jan 02 2017
%p A176479 a:= proc(n) option remember; `if`(n<2, n+1,
%p A176479      (6*n-3)/n*a(n-1) -(n-2)/(n-1)*a(n-2))
%p A176479     end:
%p A176479 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 22 2017
%t A176479 a[n_] := Sum[Binomial[n - 1, k - 1]*Binomial[n + k, n], {k, 0, n}]; Array[a, 25, 0] (* or *)
%t A176479 CoefficientList[ Series[1/4 - (x - 3)/(4 Sqrt[x^2 - 6x +1]), {x, 0, 25}], x] (* _Robert G. Wilson v_, Dec 31 2016 *)
%t A176479 Table[(n+1)Hypergeometric2F1[1-n, -n, 2, 2], {n,0,21}] (* _Peter Luschny_, Jan 02 2017 *)
%Y A176479 Cf. A001003, A105306.
%Y A176479 Row n=2 of A288972.
%K A176479 easy,nonn
%O A176479 0,2
%A A176479 _Paul Barry_, Apr 18 2010