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%I #27 Sep 16 2023 09:42:19
%S 1,1,1,5,10,16,45,109,222,540,1341,3065,7328,18112,43530,105390,
%T 260254,639244,1570257,3893805,9669236,24014264,59903650,149806494,
%U 374982790,940835404,2365679689,5955973237,15018854005,37935575685,95942896837,242954350457,616034170069,1563810857705,3974000543475
%N Number of ways to place non-crossing diagonals in convex (n+4)-gon so as to create no triangles or quadrilaterals.
%H Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, <a href="https://arxiv.org/abs/2303.13115">Blockwise simple permutations</a>, arXiv:2303.13115 [math.CO], 2023.
%H D. Birmajer, J. B. Gil, M. D. Weiner, <a href="http://arxiv.org/abs/1503.05242">Colored partitions of a convex polygon by noncrossing diagonals</a>, arXiv preprint arXiv:1503.05242 [math.CO], 2015.
%H Len Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/vsd3.html">Generalization and some variants</a>
%F a(n) = Sum_{j=0..(n-1)/3} binomial[n-2j-1, n-3j-1] binomial[n+3+j, n+2]/(n+3). This counts the polygon dissections above by number j of diagonals. - _David Callan_, Jul 15 2004
%e a(4)=5 because the octagon has the null placement and four ways to place a single diagonal.
%t f[x_] = InverseSeries[Series[(y - y^2 - y^4)/(1 - y), {y, 0, 38}], x];
%t CoefficientList[(f[x] - x)/x^4, x]
%t (* Second program: *)
%t a[n_] := Sum[Binomial[n-2j-1, n-3j-1] Binomial[n+3+j, n+2]/(n+3), {j, 0, (n-1)/3}]; Array[a, 35] (* _Jean-François Alcover_, Dec 08 2018, after _David Callan_ *)
%t Table[HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, 1 - n/3, 4 + n}, {2, 1/2 - n/2, 1 - n/2}, -27/4], {n, 1, 40}] (* _Vaclav Kotesovec_, Sep 16 2023 *)
%Y Cf. A046736, A049124, A003168, A054515.
%K nonn
%O 1,4
%A _Len Smiley_, Apr 08 2000
%E More terms from _Joerg Arndt_, Jan 28 2014
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