A033278 Number of diagonal dissections of an n-gon into 6 regions.

0, 132, 1287, 7007, 28028, 91728, 259896, 659736, 1534896, 3325608, 6789783, 13180167, 24496472, 43835792, 75869640, 127481640, 208606320, 333316620, 521215695, 799197399, 1203649524, 1783184480, 2601993680, 3743934480, 5317472160, 7461614160, 10352989647

Offset

7,2

Comments

Number of standard tableaux of shape (n-6,2,2,2,2,2) (n>=8). - Emeric Deutsch, May 20 2004

Number of short bushes with n+4 edges and 6 branch nodes (i. e. nodes with outdegree at least 2; a short bush is an ordered tree with no nodes of outdegree 1). Example: a(8) = 132 because the only short bushes with 12 edges and 6 branch nodes are the one-hundred-thirty-two full binary trees with 12 edges. Column 6 of A108263. - Emeric Deutsch, May 29 2005

Links

T. D. Noe, Table of n, a(n) for n = 7..1000

David Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, Vol. 105, No. 3 (1998), 256-257.

Frank R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., Vol. 204, No. 1-3 (1999), 73-112.

Formula

a(n) = binomial(n+4, 5)*binomial(n-3, 5)/6.

G.f.: z^8(132-165z+110z^2-44z^3+10z^4-z^5)/(1-z)^11. - Emeric Deutsch, May 29 2005

From Amiram Eldar, Oct 28 2025: (Start)

Sum_{n>=8} 1/a(n) = 391/45738.

Sum_{n>=8} (-1)^n/a(n) = 2560*log(2)/231 - 4914367/640332. (End)

Mathematica

a[n_] := Binomial[n+4, 5] * Binomial[n-3, 5]/6; Array[a, 20, 7] (* Amiram Eldar, Oct 28 2025 *)

Prog

(PARI) vector(40, n, n+=6; binomial(n+4, 5)*binomial(n-3, 5)/6) \\ Michel Marcus, Jun 18 2015

Crossrefs

Cf. A108263.

Sequence in context: A305271 A090199 A239598 * A362104 A213380 A119982

Adjacent sequences: A033275 A033276 A033277 * A033279 A033280 A033281

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved