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

Author

Michael D. Weiner, Mar 24 2015

Keywords

• nonn

Examples

			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.
		

Links

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

Crossrefs

Cf. A255197.

Programs

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 *)

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