The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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