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 A350303 a(n) is the number of ways to partition the set of vertices of a convex (n+14)-gon into 5 nonintersecting polygons. 1
 0, 273, 1820, 7140, 21420, 54264, 122094, 251370, 482790, 876645, 1519518, 2532530, 4081350, 6388200, 9746100, 14535612, 21244356, 30489585, 43044120, 59865960, 82131896, 111275472, 149029650, 197474550, 259090650, 336817845, 434120778, 555060870, 704375490, 887564720, 1110986184 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Equivalently, the number of noncrossing set partitions of an (n+14)-set into 5 blocks with 3 or more elements in each block. LINKS Table of n, a(n) for n=0..30. Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1). FORMULA a(n) = (1/2880)*n*(n+1)*(n+2)*(n+3)*(n+11)*(n+12)*(n+13)*(n+14). G.f.: 7*x*(39 - 91*x + 84*x^2 - 36*x^3 + 6*x^4)/(1 - x)^9. - Stefano Spezia, Dec 26 2021 EXAMPLE The a(1)=273 solutions are {1,2,3} {4,5,6} {7,8,9} {10,11,12} {13,14,15} with its 3 different orientations and each of the following 18 patterns with its 15 orientations: {1,2,3} {4,5,15} {6,7,8} {9,10,11} {12,13,14} {1,2,3} {4,14,15} {5,6,7} {8,9,10} {11,12 13} {1,2,3} {4,5,6} {7,8,15} {9,10,11} {12 13,14} {1,2,3} {4,5,15} {6,7,14} {8,9,10} {11,12,13} {1,2,3} {4,14,15} {5,12,13} {6,7,8} {9,10,11} {1,2,3} {4,5,15} {6,13,14} {7,8,9} {10,11,12} {1,2,3} {4,14,15} {5,6,13} {7,8,9} {10,11,12} {1,2,3} {4,5,15} {6,7,14} {8,9,13} {10,11,12} {1,2,3} {4,5,15} {6,7,14} {8,12,13} {9,10,11} {1,2,3} {4,5,15} {6,13,14} {7,8,12} {9,10,11} {1,2,3} {4,14,15} {5,12,13} {6,7,11} {8,9,10} {1,2,3} {4,15,8} {5,6,7} {9,10,11} {12,13,14} {1,2,3} {4,15,8} {5,6,7} {9,13,14} {10,11,12} {1,2,3} {4,15,8} {5,6,7} {9,10,14} {11,12,13} {1,2,3} {4,5,15} {6,7,8} {9,10,14} {11,12,13} {1,2,3} {4,14,15} {5,6,7} {8,12,13} {9,10,11} {1,2,3} {4,14,15} {5,6,7} {8,9,13} {10,11,12} {1,2,3} {4,5,15} {6,7,8} {9,13,14} {10,11,12} In the above, the numbers can be considered to be the partition of a 15-set into 5 blocks or the partition of the vertices of a convex 15-gon into 5 triangles with vertices labeled 1,2,...,15 in order. a(2)=1820 corresponding to the number of ways to partition the vertices of a 16-gon into 4 triangles and one quadrilateral. MATHEMATICA a[n_] := n*(n + 1)*(n + 2)*(n + 3)*(n + 11)*(n + 12)*(n + 13)*(n + 14)/2880; Array[a, 30, 0] (* Amiram Eldar, Dec 26 2021 *) CROSSREFS Column k=5 of A350248. Cf. A350116. Sequence in context: A028530 A200836 A276980 * A214220 A029567 A028534 Adjacent sequences: A350300 A350301 A350302 * A350304 A350305 A350306 KEYWORD easy,nonn AUTHOR Janaka Rodrigo, Dec 24 2021 STATUS approved

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Last modified September 22 08:45 EDT 2023. Contains 365519 sequences. (Running on oeis4.)