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A002296
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Number of dissections of a polygon: binomial(7n,n)/(6n+1).
(Formerly M4442 N1878)
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48
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1, 1, 7, 70, 819, 10472, 141778, 1997688, 28989675, 430321633, 6503352856, 99726673130, 1547847846090, 24269405074740, 383846168712104, 6116574500860880, 98106248306858715, 1582638261961640247, 25661404527790252375, 417980115131315136400
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OFFSET
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0,3
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COMMENTS
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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 Pólya-Szegő reference.
Also 7-Raney sequence. See the Graham et al. reference, pp. 346-347.
This is instance k = 7 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564. - Wolfdieter Lang, Feb 05 2024
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nürnberg, Jul 27 1994.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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.
D-finite with recurrence: 72*n*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*(6*n+1)*a(n) - 7*(7*n-3)*(7*n-6)*(7*n-2)*(7*n-5)*(7*n-1)*(7*n-4)*a(n-1) = 0. - R. J. Mathar, Nov 16 2012
a(n) are special values of Jacobi polynomials, in Maple notation:
a(n) = JacobiP(n-1, 6*n+1, -n, 1)/n, n = 1, 2, ... . - Karol A. Penson, Mar 16 2015
a(0) = 1; a(n) = Sum_{i1+i2+...+i7=n-1} a(i1)*a(i2)*...*a(i7) for n>=1. - Robert FERREOL, Apr 02 2015
O.g.f.: 6F5(1/7,2/7,3/7,4/7,5/7,6/7; 1/3,1/2,2/3,5/6,7/6; 823543*x/46656).
E.g.f.: 6F6(1/7,2/7,3/7,4/7,5/7,6/7; 1/3,1/2,2/3,5/6,1,7/6; 823543*x/46656).
a(n) ~ 7^(7*n+1/2)/(sqrt(Pi)*3^(6*n+3/2)*4^(3*n+1)*n^(3/2)). (End)
x*A'(x)/A(x) = (A(x) - 1)/(- 6*A(x) + 7) = x + 13*x^2 + 190*x^3 + 2925*x^4 + ... = (1/7)*Sum_{n >= 1} binomial(7*n,n)*x^n. Cf. A001764 and A002293, A002294, A002295. - Peter Bala, Feb 04 2022
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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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MAPLE
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seq(binomial(7*n+1, n)/(7*n+1), n=0..30); # Robert FERREOL, Apr 02 2015
n:=30: G:=series(RootOf(g = 1+x*g^7, g), x=0, n+1): seq(coeff(G, x, k), k=0..n); # Robert FERREOL, Apr 02 2015
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MATHEMATICA
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Table[Binomial[7n, n]/(6n+1), {n, 0, 20}] (* Harvey P. Dale, Nov 21 2011 *)
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PROG
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(Haskell)
a002296 n = a002296_list !! n
a002296_list = [a258708 (4 * n) (3 * n) | n <- [1..]]
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CROSSREFS
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KEYWORD
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easy,nonn,nice
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AUTHOR
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EXTENSIONS
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Pfaff-Fuss-Catalan, Raney, o.g.f. and 7-ary tree comments from Wolfdieter Lang, Sep 14 2007
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STATUS
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approved
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