A054514 Number of ways to place non-crossing diagonals in convex (n+4)-gon so as to create no triangles or quadrilaterals.

1, 1, 1, 5, 10, 16, 45, 109, 222, 540, 1341, 3065, 7328, 18112, 43530, 105390, 260254, 639244, 1570257, 3893805, 9669236, 24014264, 59903650, 149806494, 374982790, 940835404, 2365679689, 5955973237, 15018854005, 37935575685, 95942896837, 242954350457, 616034170069, 1563810857705, 3974000543475

Offset

1,4

Links

Table of n, a(n) for n=1..35.

Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Blockwise simple permutations, arXiv:2303.13115 [math.CO], 2023.

Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Geometric view of interval poset permutations, arXiv:2411.13193 [math.CO], 2024. See p. 10.

D. Birmajer, J. B. Gil, M. D. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv preprint arXiv:1503.05242 [math.CO], 2015.

Len Smiley, Generalization and some variants

Formula

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

Example

a(4)=5 because the octagon has the null placement and four ways to place a single diagonal.

Mathematica

f[x_] = InverseSeries[Series[(y - y^2 - y^4)/(1 - y), {y, 0, 38}], x];
CoefficientList[(f[x] - x)/x^4, x]
(* Second program: *)
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 *)
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 *)

Crossrefs

Cf. A046736, A049124, A003168, A054515.

Sequence in context: A301290 A152234 A357997 * A372624 A200940 A002660

Adjacent sequences: A054511 A054512 A054513 * A054515 A054516 A054517

Keyword

nonn

Author

Len Smiley, Apr 08 2000

Extensions

More terms from Joerg Arndt, Jan 28 2014

Status

approved