cp's OEIS Frontend

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

%I A253192 #30 Nov 28 2024 18:56:18
%S A253192 1,0,5,6,35,80,309,890,3058,9580,31863,103054,340415,1116032,3688745,
%T A253192 12176814,40344505,133742500,444262378,1477142040,4918099660,
%U A253192 16390294664,54679621775,182572812266,610115196150,2040383498748,6828408179435,22866979920390,76623655367703,256899191586880,861774049296325
%N A253192 Number of ways to place nonintersecting diagonals in convex (n+3)-gon so as to create exactly one triangle.
%H A253192 D. Birmajer, J. B. Gil, and M. Weiner, <a href="http://arxiv.org/abs/1503.05242">Colored partitions of a convex polygon by noncrossing diagonals</a>, arXiv:1503.05242 [math.CO], 2015.
%F A253192 a(n) = Sum_{k=1..floor(n/2)} C(n+k+2,k)*C(n-k-1,k-1), n>0.
%F A253192 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).
%F A253192 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
%e A253192 a(1)=0 since there are no dissections of a convex quadrilateral with exactly one triangle.
%e A253192 a(2)=5 because we can place one diagonal in a pentagon 5 different ways, each time creating one triangle and one quadrilateral.
%p A253192 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));
%p A253192 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);
%p A253192 gfun[seriestolist](series(ogf, x=0, 30))[]; # _Mark van Hoeij_, Nov 28 2024
%t A253192 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 *)
%Y A253192 Cf. A255197.
%K A253192 nonn
%O A253192 0,3
%A A253192 _Michael D. Weiner_, Mar 24 2015