A253192 Number of ways to place nonintersecting diagonals in convex (n+3)-gon so as to create exactly one triangle.

1, 0, 5, 6, 35, 80, 309, 890, 3058, 9580, 31863, 103054, 340415, 1116032, 3688745, 12176814, 40344505, 133742500, 444262378, 1477142040, 4918099660, 16390294664, 54679621775, 182572812266, 610115196150, 2040383498748, 6828408179435, 22866979920390, 76623655367703, 256899191586880, 861774049296325

Offset

0,3

Links

Table of n, a(n) for n=0..30.

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

Formula

a(n) = Sum_{k=1..floor(n/2)} C(n+k+2,k)*C(n-k-1,k-1), n>0.

D-finite with recurrence: 0=2*(n-1)*(2*n-3)*(n+1)*(37*n^3 + 97*n^2 + 76*n + 20)*a(n-3) - 2*n*(592*n^5 + 960*n^4 - 15*n^3 - 70*n^2 + 263*n + 70)*a(n-2) - 2*n*(n-1)*(n+1)*(148*n^3 + 314*n^2 + 37*n - 89)*a(n-1) + 5*n*(n+2)*(n+1)*(37*n^3 - 14*n^2 - 7*n + 4)*a(n).

a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 3.4086981998421510858648764973336... is the real root of the equation 4 - 32*d - 8*d^2 + 5*d^3 = 0 and c = 0.8203071528123829561131676776610304796... is the smallest positive real root of the equation 1 + 402019*c - 584933*c^2 + 115625*c^3 = 0. - Vaclav Kotesovec, Jul 05 2024

Example

a(1)=0 since there are no dissections of a convex quadrilateral with exactly one triangle.
a(2)=5 because we can place one diagonal in a pentagon 5 different ways, each time creating one triangle and one quadrilateral.

Maple

a:=n->sum(binomial(n+k+2, k)*binomial(n-k-1, k-1), k = 1 .. trunc((1/2)*n)): (1, seq(a(n), n=1..30));
ogf := (RootOf((4*x^3-32*x^2-8*x+5)*_Z^3+(-9*x^4+42*x^3+249*x^2+96*x-51)*_Z+18*x^4-116*x^3-269*x^2-128*x+62)-2)/(3*x^2);
gfun[seriestolist](series(ogf, x=0, 30))[]; # Mark van Hoeij, Nov 28 2024

Mathematica

Prepend[Table[Sum[Binomial[n + k + 2, k]*Binomial[n - k - 1, k - 1], {k, 1, n/2}], {n, 1, 30}], 1] (* Michael De Vlieger, Mar 24 2015 *)

Crossrefs

Cf. A255197.

Sequence in context: A248254 A212918 A255197 * A036254 A047170 A333218

Adjacent sequences: A253189 A253190 A253191 * A253193 A253194 A253195

Keyword

nonn

Author

Michael D. Weiner, Mar 24 2015

Status

approved