OFFSET
1,4
LINKS
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
KEYWORD
nonn
AUTHOR
Len Smiley, Apr 08 2000
EXTENSIONS
More terms from Joerg Arndt, Jan 28 2014
STATUS
approved