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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-Fran├žois Alcover, Jul 29 2018 *)

CROSSREFS

Cf. A000108, A002054, A002694, A046736.

Sequence in context: A324185 A175958 A021469 * A011131 A330602 A058241

Adjacent sequences:  A090982 A090983 A090984 * A090986 A090987 A090988

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Feb 28 2004

STATUS

approved

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Last modified August 9 05:07 EDT 2020. Contains 336319 sequences. (Running on oeis4.)