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A089382 Total number of triangles in all the dissections of a convex (n+3)-gon by nonintersecting diagonals. 1
1, 4, 20, 104, 553, 2984, 16272, 89440, 494681, 2749772, 15348372, 85967112, 482927985, 2719787856, 15351385152, 86816721792, 491819758417, 2790451952660, 15854070902964, 90187514559208, 513619224125657, 2928073006131704, 16708228671139600, 95423104768226144, 545408567460801513 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

G.f.: (3-z+q)*(1+z-q)^2/(64*q*z^2), where q = sqrt(1-6*z+z^2).

Recurrence: n*(n+2)*a(n) = (8*n^2 + 7*n - 3)*a(n-1) - (13*n^2 - 17*n - 6)*a(n-2) + 2*(n-3)*(n+1)*a(n-3). - Vaclav Kotesovec, Oct 14 2012

a(n) ~ sqrt(140+99*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

a(n) = Sum_{j=0..n}((-1)^j*2^(n-j)*binomial(n+3,j)*binomial(2*n-j+2,n+2)). - Vladimir Kruchinin, Apr 08 2016

EXAMPLE

a(1)=4 because in the three dissections of a square we have altogether four triangles: no triangle in the "no-diagonals" dissection and two triangles in each of the dissections by one of the two diagonals of the square.

MATHEMATICA

Table[SeriesCoefficient[(3-x+Sqrt[1-6*x+x^2])*(1+x-Sqrt[1-6*x+x^2])^2/(64*Sqrt[1-6*x+x^2]*x^2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)

PROG

(PARI)  x='x+O('x^66); q = sqrt(1-6*x+x^2); Vec((3-x+q)*(1+x-q)^2/(64*q*x^2)) \\ Joerg Arndt, May 10 2013

(Maxima)

a(n):=sum((-1)^j*2^(n-j)*binomial(n+3, j)*binomial(2*n-j+2, n+2), j, 0, n); /* Vladimir Kruchinin, Apr 08 2016 */

CROSSREFS

Cf. A001003.

Sequence in context: A120978 A035028 A104550 * A192619 A026305 A195256

Adjacent sequences:  A089379 A089380 A089381 * A089383 A089384 A089385

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 28 2003

STATUS

approved

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Last modified June 28 20:39 EDT 2017. Contains 288840 sequences.