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A350116
Number of ways to partition the set of vertices of a convex {n+8}-gon into 3 non-intersecting polygons.
8
0, 12, 45, 110, 220, 390, 637, 980, 1440, 2040, 2805, 3762, 4940, 6370, 8085, 10120, 12512, 15300, 18525, 22230, 26460, 31262, 36685, 42780, 49600, 57200, 65637, 74970, 85260, 96570, 108965, 122512, 137280, 153340, 170765, 189630, 210012, 231990, 255645, 281060, 308320
OFFSET
0,2
COMMENTS
Equivalently, the number of noncrossing set partitions of an {n+8}-set into 3 blocks with 3 or more elements in each block.
FORMULA
a(n) = n*(n+1)*(n+7)*(n+8)/12.
G.f.: -x*(12-15*x+5*x^2)/(x-1)^5 . - R. J. Mathar, Aug 03 2022
EXAMPLE
The a(1) = 12 solutions are:
{123}{456}{789}, {234}{567}{891}, {345}{678}{912},
{156}{234}{567}, {267}{345}{891}, {378}{456}{912},
{489}{567}{123}, {591}{678}{234}, {612}{789}{345},
{723}{891}{456}, {834}{912}{567}, {945}{123}{678}.
In the above, the numbers can be considered to be the partition of a 9-set into 3 blocks or the partition of the vertices of a convex 9-gon into 3 triangles (with the vertices labeled 1..9 in order).
a(2) = 45 corresponding to the number of ways to partition the vertices of a 10-gon into two triangles and one quadrilateral.
MATHEMATICA
a[n_] := n*(n + 1)*(n + 7)*(n + 8)/12; Array[a, 40, 0] (* Amiram Eldar, Dec 21 2021 *)
CROSSREFS
Column k=3 of A350248.
The case of any number of parts for an n-gon is A114997.
The case of exactly 2 parts for a {n+5}-gon is A055998.
Sequence in context: A135710 A358465 A070996 * A015237 A024223 A251720
KEYWORD
easy,nonn
AUTHOR
Janaka Rodrigo, Dec 21 2021
STATUS
approved