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A100299 Number of dissections of a convex n-gon by nonintersecting diagonals into an even number of regions. 2
0, 2, 5, 23, 98, 452, 2139, 10397, 51524, 259430, 1323361, 6824435, 35519686, 186346760, 984400759, 5231789177, 27954506504, 150079713482, 809181079293, 4379654830223, 23787413800490, 129607968854732, 708230837732435, 3880366912218773, 21312485647242828, 117321536967959342 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 3..200

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.

FORMULA

a(n) = sum(C(n-3, 2k-1)*C(n+2k-2, 2k-1)/(2k), k=1..floor((n-2)/2)).

G.f.: (1/2)*z^2/(1+z)+z/8-7*z^2/8-(1/8)*z*sqrt(1-6*z+z^2).

Recurrence (for n>4): (n-1)*(2*n-7)*a(n) = (2*n-5)*(5*n-19)*a(n-1)+(5*n-11)*(2*n-7)*a(n-2)-(2*n-5)*(n-5)*a(n-3). - Vaclav Kotesovec, Oct 17 2012

Asymptotic: a(n) ~ sqrt(3*sqrt(2)-4)*(3+2*sqrt(2))^(n-1) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

D-finite with recurrence (-n+1)*a(n) +(4*n-11)*a(n-1) +5*(2*n-7)*a(n-2) +(4*n-17)*a(n-3) +(-n+6)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

EXAMPLE

a(5)=5 because for a convex pentagon ABCDE we obtain dissections with an even number of regions by one of the following sets of diagonals: {AC}, {BD}, {CE}, {DA} and {EB}.

MAPLE

a:=n->sum(binomial(n-3, 2*k-1)*binomial(n+2*k-2, 2*k-1)/2/k, k=1..floor((n-2)/2)): seq(a(n), n=3..33);

MATHEMATICA

Take[CoefficientList[Series[1/2*x^2/(1+x)+x/8-7*x^2/8-x/8*Sqrt[1-6*x+x^2], {x, 0, 20}], x], {4, -1}] (* Vaclav Kotesovec, Oct 17 2012 *)

PROG

(PARI) x='x+O('x^66); concat([0], Vec((1/2)*x^2/(1+x)+x/8-7*x^2/8-(1/8)*x*sqrt(1-6*x+x^2))) \\ Joerg Arndt, May 12 2013

(PARI) a(n) = sum(k=1, (n-2)\2, binomial(n-3, 2*k-1)*binomial(n+2*k-2, 2*k-1)/(2*k)); \\ Altug Alkan, Oct 26 2015

CROSSREFS

Cf. A100300.

Sequence in context: A106858 A290887 A219889 * A038833 A279819 A249606

Adjacent sequences: A100296 A100297 A100298 * A100300 A100301 A100302

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 12 2004

STATUS

approved

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Last modified November 30 02:32 EST 2022. Contains 358431 sequences. (Running on oeis4.)