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

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A090985 Triangle read by rows: T(n,k) is the number of dissections of a convex n-gon by nonintersecting diagonals, having exactly k triangles (n >= 2, k >= 0). 0
 1, 0, 1, 1, 0, 2, 1, 5, 0, 5, 4, 6, 21, 0, 14, 8, 35, 28, 84, 0, 42, 25, 80, 216, 120, 330, 0, 132, 64, 309, 540, 1155, 495, 1287, 0, 429, 191, 890, 2475, 3080, 5720, 2002, 5005, 0, 1430, 540, 3058, 7788, 16302, 16016, 27027, 8008, 19448, 0, 4862, 1616, 9580, 30108, 54964, 96005, 78624, 123760, 31824, 75582, 0, 16796 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,6 COMMENTS T(n,n-2) = [binomial(2n-4, n-2)]/(n-1) = Catalan(n-2) (A000108). T(n,n-4) = binomial(2n-5, n-4) (A002054). T(n,n-5) = binomial(2n-6, n-5) (A002694). T(n,0) = A046736(n). Row sums give the little Schroeder numbers (A001003). LINKS Table of n, a(n) for n=2..67. P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999. FORMULA T(n, k) = binomial(n+k-2, k)*(Sum_{i=0..floor((n-2-k)/2)} binomial(n-2+k+i, i)*binomial(n-3-k-i, i-1))/(n-1). G.f.: G=G(t, z) satisfies (1-t)G^3 + (1+t)zG^2 - z^2*(1+z)G + z^4 = 0. EXAMPLE T(5,1)=5 because the dissections of a convex pentagon having exactly one triangle are obtained by the placement of a diagonal between any pair of non-adjacent vertices. T(6,0)=4 because the dissections of a convex hexagon with no triangles are obtained by the null placement and by placing one diagonal between any of the 3 pairs of opposite vertices. Triangle starts: 1; 0, 1; 1, 0, 2; 1, 5, 0, 5; 4, 6, 21, 0, 14; 8, 35, 28, 84, 0, 42; ... MAPLE T := (n, k)->binomial(n+k-2, k)*sum(binomial(n-2+k+i, i)*binomial(n-3-k-i, i-1), i=0..floor((n-2-k)/2))/(n-1): seq(seq(T(n, k), k=0..n-2), n=2..14); MATHEMATICA T [n_, k_] := Binomial[n+k-2, k] Sum[Binomial[n-2+k+i, i] Binomial[n-3-k-i, i-1], {i, 0, (n-2-k)/2}]/(n-1); Table[T[n, k], {n, 2, 12}, {k, 0, n-2}] // Flatten (* Jean-François Alcover, Jul 29 2018 *) CROSSREFS Cf. A000108, A002054, A002694, A046736. Sequence in context: A348175 A175958 A021469 * A011131 A330602 A058241 Adjacent sequences: A090982 A090983 A090984 * A090986 A090987 A090988 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Feb 28 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 July 22 22:22 EDT 2024. Contains 374544 sequences. (Running on oeis4.)