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Search: id:A002296
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| A002296 |
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Number of dissections of a polygon: C(7n,n)/(6n+1).
(Formerly M4442 N1878)
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+0
9
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| 1, 1, 7, 70, 819, 10472, 141778, 1997688, 28989675, 430321633, 6503352856, 99726673130, 1547847846090, 24269405074740, 383846168712104, 6116574500860880, 98106248306858715, 1582638261961640247, 25661404527790252375, 417980115131315136400
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a(n), n>=1, enumerates heptic (7-ary) trees (rooted, ordered, incomplete) with n vertices (including the root).
Pfaff-Fuss-Catalan sequence C^{m}_n for m=7. See the Graham et al. reference, p. 347. eq. 7.66. See also the P\'olya-Szeg\"o reference.
Also 7-Raney sequence. See the Graham et al. reference, p. 346-7.
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REFERENCES
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F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).
Editor's note: "Ueber die Bestimmung der Anzahl der verschiedenen Arten, auf welche sich ein n-Eck durch Diagonalen in lauter m-Ecke zerlegen laesst, mit Bezug auf einige Abhandlungen der Herren Lame, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathematiques pure et appliquees, publie par Joseph Liouville. T. III. IV.", Archiv der Mathematik u. Physik, 1 (1841), pp. 193ff; see especially p. 198.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
G. P\'olya and G. Szeg\"o, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 289
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FORMULA
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O.g.f. A(x)= 1 + x*A(x)^7 = 1/(1-x*A(x)^6).
a(n)=binomial(7*n,n-1)/n, n>=1, a(0)=1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.
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EXAMPLE
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There are a(2)=7 heptic trees (vertex degree <=7 and 7 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 7 trees yields 7*7+binomial(7,2)=70=a(3) such trees.
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CROSSREFS
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Cf. A002294, A002296.
Sixth column of triangle A062993.
Adjacent sequences: A002293 A002294 A002295 this_sequence A002297 A002298 A002299
Sequence in context: A015251 A078246 A097184 this_sequence A027394 A113343 A124566
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KEYWORD
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easy,nonn,nice
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AUTHOR
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njas
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EXTENSIONS
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Pfaff-Fuss-Catalan, Raney, o.g.f. and 7-ary tree comments from W. Lang, Sep 14 2007.
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