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%I M4442 N1878 #114 Feb 05 2024 08:26:05
%S 1,1,7,70,819,10472,141778,1997688,28989675,430321633,6503352856,
%T 99726673130,1547847846090,24269405074740,383846168712104,
%U 6116574500860880,98106248306858715,1582638261961640247,25661404527790252375,417980115131315136400
%N Number of dissections of a polygon: binomial(7n,n)/(6n+1).
%C a(n), n>=1, enumerates heptic (7-ary) trees (rooted, ordered, incomplete) with n vertices (including the root).
%C 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.
%C Also 7-Raney sequence. See the Graham et al. reference, pp. 346-347.
%C a(n) = A258708(3*n,2*n) for n > 0. - _Reinhard Zumkeller_, Jun 23 2015
%C 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
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
%D 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.
%D Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nürnberg, Jul 27 1994.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002296/b002296.txt">Table of n, a(n) for n = 0..100</a>
%H Wun-Seng Chou, Tian-Xiao He, Peter J.-S. Shiue, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/He/he61.html">On the Primality of the Generalized Fuss-Catalan Numbers</a>, J. Int. Seqs., Vol. 21 (2018), #18.2.1.
%H F. Harary, E. M. Palmer, and R. C. Read, <a href="/A000108/a000108_20.pdf">On the cell-growth problem for arbitrary polygons</a>, computer printout, circa 1974.
%H F. Harary, E. M. Palmer and R. C. Read, <a href="http://dx.doi.org/10.1016/0012-365X(75)90041-2">On the cell-growth problem for arbitrary polygons</a>, Discr. Math. 11 (1975), 371-389. See Table 6. n = 8. Sequence U(8) p. 387.
%H Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, <a href="https://arxiv.org/abs/2204.14023">Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k</a>, arXiv:2204.14023 [math.CO], 2022.
%H V. E. Hoggatt, Jr., <a href="/A005676/a005676.pdf">7-page typed letter to N. J. A. Sloane with suggestions for new sequences</a>, circa 1977.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=289">Encyclopedia of Combinatorial Structures 289</a>.
%H R. P. Loh, A. G. Shannon, and A. F. Horadam, <a href="/A000969/a000969.pdf">Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients</a>, Preprint, 1980.
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv:1403.5962 [math.CO], 2014.
%H Editor's note: <a href="http://books.google.com/books?id=oVvxAAAAMAAJ">Über 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 pures et appliquees, publie par Joseph Liouville. T. III. IV.</a>, Archiv der Mathematik u. Physik, 1 (1841), pp. 193ff; see especially p. 198.
%H B. Sury, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Sury/sury31.html">Generalized Catalan numbers: linear recursion and divisibility</a>, JIS 12 (2009), Article 09.7.5.
%H Lajos Takacs, <a href="http://www.appliedprobability.org/data/files/TMS%20articles/18_1_1.pdf">Enumeration of rooted trees and forests</a>, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).
%F O.g.f. A(x) = 1 + x*A(x)^7 = 1/(1-x*A(x)^6).
%F 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.
%F 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
%F a(n) are special values of Jacobi polynomials, in Maple notation:
%F a(n) = JacobiP(n-1, 6*n+1, -n, 1)/n, n = 1, 2, ... . - _Karol A. Penson_, Mar 16 2015
%F a(n) = binomial(7*n+1, n)/(7*n+1) = A062993(n+5,5). - _Robert FERREOL_, Apr 02 2015
%F 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
%F From _Ilya Gutkovskiy_, Jan 16 2017: (Start)
%F 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).
%F 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).
%F a(n) ~ 7^(7*n+1/2)/(sqrt(Pi)*3^(6*n+3/2)*4^(3*n+1)*n^(3/2)). (End)
%F 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
%e 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.
%p seq(binomial(7*n+1, n)/(7*n+1), n=0..30); # _Robert FERREOL_, Apr 02 2015
%p 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
%t Table[Binomial[7n,n]/(6n+1),{n,0,20}] (* _Harvey P. Dale_, Nov 21 2011 *)
%o (PARI) a(n)=binomial(7*n,n)/(6*n+1) \\ _Charles R Greathouse IV_, Feb 06 2012
%o (Haskell)
%o a002296 n = a002296_list !! n
%o a002296_list = [a258708 (4 * n) (3 * n) | n <- [1..]]
%o -- _Reinhard Zumkeller_, Jun 23 2015
%Y Cf. A001764, A002293, A002294, A002295.
%Y Sixth column of triangle A062993.
%Y Cf. A235535: binomial(9n,3n)/(6n+1); A235536: binomial(8n,2n)/(6n+1).
%Y Cf. A258708.
%Y Cf. A130564.
%K easy,nonn,nice
%O 0,3
%A _N. J. A. Sloane_
%E Pfaff-Fuss-Catalan, Raney, o.g.f. and 7-ary tree comments from _Wolfdieter Lang_, Sep 14 2007
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