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a(n) = (n+1)*(3*n-2)*C(2*n,n-1)/(4*n-2).
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%I #24 Apr 01 2022 09:06:19

%S 1,8,42,200,910,4032,17556,75504,321750,1361360,5727436,23984688,

%T 100053772,416024000,1725013800,7135405920,29452939110,121347523440,

%U 499132441500,2050025300400,8408638258020,34448503964160,140974630569240,576340150932000,2354075068866300

%N a(n) = (n+1)*(3*n-2)*C(2*n,n-1)/(4*n-2).

%C This is half the number of edges of the origami flip graph of the all-equal-angle single-vertex crease pattern of degree 2n.

%H Michael De Vlieger, <a href="/A352626/b352626.txt">Table of n, a(n) for n = 1..1658</a>

%H Thomas C. Hull, Manuel Morales, Sarah Nash, and Natalya Ter-Saakov, <a href="https://arxiv.org/abs/2203.14173">Maximal origami flip graphs of flat-foldable vertices: properties and algorithms</a>, arXiv:2203.14173 [math.CO], 2022.

%F G.f.: x*(2*x+1)/(1-4*x)^(3/2).

%F D-finite with recurrence (-n+1)*a(n) +2*(n+2)*a(n-1) +4*(2*n-5)*a(n-2)=0. - _R. J. Mathar_, Mar 29 2022

%F a(n) = A002457(n-1)+2*A002457(n-2). - _R. J. Mathar_, Mar 29 2022

%p a:=n->(n+1)*(3*n-2)*binomial(2*n,n-1)/(4*n-2):

%p seq(a(n),n=1..33);

%t a[n_] := (n + 1)*(3*n - 2)*Binomial[2*n, n - 1]/(4*n - 2); Array[a, 25] (* _Amiram Eldar_, Mar 25 2022 *)

%o (PARI) a(n) = (n+1)*(3*n-2)*binomial(2*n,n-1)/(4*n-2); \\ _Michel Marcus_, Mar 25 2022

%Y Number of vertices of the origami flip graph of the all-equal-angle single-vertex crease pattern of degree 2n is A162551.

%K nonn,easy

%O 1,2

%A _Natalya Ter-Saakov_, Mar 24 2022