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A253192
Number of ways to place nonintersecting diagonals in convex (n+3)-gon so as to create exactly one triangle.
1
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
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..n/2} C(n+k+2,k)*C(n-k-1,k-1), n>1.
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));
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
KEYWORD
nonn
AUTHOR
Michael D. Weiner, Mar 24 2015
STATUS
approved