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A002295 Number of dissections of a polygon: binomial(6n,n)/(5n+1).

%I A002295 M4260 N1780 #156 Jun 16 2025 08:47:31
%S A002295 1,1,6,51,506,5481,62832,749398,9203634,115607310,1478314266,
%T A002295 19180049928,251857119696,3340843549855,44700485049720,
%U A002295 602574657427116,8175951659117794,111572030260242090,1530312970340384580,21085148778264281865,291705220704719165526
%N A002295 Number of dissections of a polygon: binomial(6n,n)/(5n+1).
%C A002295 From _Wolfdieter Lang_, Sep 14 2007: (Start)
%C A002295 a(n), n >= 1, enumerates sextic (6-ary) trees (rooted, ordered, incomplete) with n vertices (including the root).
%C A002295 Pfaff-Fuss-Catalan sequence C^{m}_n for m=6. See the Graham et al. reference, p. 347. eq. 7.66. See also the Pólya-Szegő reference.
%C A002295 Also 6-Raney sequence. See the Graham et al. reference, p. 346-7. (End)
%C A002295 This is instance k = 6 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564. - _Wolfdieter Lang_, Feb 05 2024
%D A002295 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 23.
%D A002295 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
%D A002295 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 A002295 Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nürnberg, Jul 27 1994
%D A002295 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002295 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002295 Editor's note: "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 Lamé, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathématiques pures et appliquées, publié par Joseph Liouville. T. III. IV.", Archiv der Mathematik u. Physik, 1 (1841), pp. 193ff; see especially p. 198.
%H A002295 T. D. Noe, <a href="/A002295/b002295.txt">Table of n, a(n) for n = 0..100</a>
%H A002295 V. E. Adler and A. B. Shabat, <a href="https://arxiv.org/abs/1810.13198">Volterra chain and Catalan numbers</a>, arXiv:1810.13198 [nlin.SI], 2018.
%H A002295 Paul Barry, <a href="https://arxiv.org/abs/2104.05593">On the Gap-sum and Gap-product Sequences of Integer Sequences</a>, arXiv:2104.05593 [math.CO], 2021.
%H A002295 Wun-Seng Chou, Tian-Xiao He, and 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 A002295 Olivier Danvy, <a href="https://arxiv.org/abs/2412.03127">Summa Summarum: Moessner's Theorem without Dynamic Programming</a>, arXiv:2412.03127 [cs.DM], 2024. See p. 25.
%H A002295 F. Harary, E. M. Palmer, and R. C. Read, <a href="/A000108/a000108_20.pdf">On the cell-growth problem for arbitrary polygons, computer printout, circa 1974</a>
%H A002295 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.
%H A002295 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 A002295 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 A002295 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=288">Encyclopedia of Combinatorial Structures 288</a>
%H A002295 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 A002295 J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
%H A002295 Yuan (Friedrich) Qiu, Joe Sawada, and Aaron Williams, <a href="https://www.socs.uoguelph.ca/~sawada/papers/RSG.pdf">Maximize the Rightmost Digit: Gray Codes for Restricted Growth Strings</a>, WALCOM 2025. See p. 5.
%H A002295 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 A002295 L. Takacs, <a href="https://web.archive.org/web/20201117053023/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).
%H A002295 Jun Yan, <a href="https://arxiv.org/abs/2501.01152">Lattice paths enumerations weighted by ascent lengths</a>, arXiv:2501.01152 [math.CO], 2025. See p. 7.
%F A002295 O.g.f.: A(x) = 1 + x*A(x)^6 = 1/(1-x*A(x)^5).
%F A002295 a(n) = binomial(6*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 A002295 a(n) = upper left term in M^n, M = the production matrix:
%F A002295    1,  1
%F A002295    5,  5,  1
%F A002295   15, 15,  5,  1
%F A002295   35, 35, 15,  5,  1
%F A002295   ...
%F A002295 where (1, 5, 15, 35, ...) = A000332 starting with 1. - _Gary W. Adamson_, Jul 08 2011
%F A002295 a(n) are special values of Jacobi polynomials, in Maple notation:
%F A002295   a(n) = JacobiP(n-1, 5*n+1, -n, 1)/n, n=1, 2, ... . - _Karol A. Penson_, Mar 17 2015
%F A002295 a(n) = binomial(6*n+1, n)/(6*n+1) = A062993(n+4,4). - _Robert FERREOL_, Apr 03 2015
%F A002295 a(0) = 1; a(n) = Sum_{i1+i2+...+i6=n-1} a(i1)*a(i2)*...*a(i6) for n>=1. - _Robert FERREOL_, Apr 03 2015
%F A002295 D-finite with recurrence: 5*n*(5*n+1)*(5*n-3)*(5*n-2)*(5*n-1)*a(n) - 72*(6*n-5)*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*a(n-1) = 0. - _R. J. Mathar_, Sep 06 2016
%F A002295 From _Ilya Gutkovskiy_, Jan 15 2017: (Start)
%F A002295 O.g.f.: 5F4(1/6,1/3,1/2,2/3,5/6; 2/5,3/5,4/5,6/5; 46656*x/3125).
%F A002295 E.g.f.: 5F5(1/6,1/3,1/2,2/3,5/6; 2/5,3/5,4/5,1,6/5; 46656*x/3125).
%F A002295 a(n) ~ 3^(6*n+1/2)*64^n/(sqrt(Pi)*5^(5*n+3/2)*n^(3/2)). (End)
%F A002295 x*A'(x)/A(x) = (A(x) - 1)/(- 5*A(x) + 6) = x + 11*x^2 + 136*x^3 + 1771*x^4 + ... = (1/6)*Sum_{n >= 1} binomial(6*n,n)*x^n. Cf. A001764 and A002293 - A002296. - _Peter Bala_, Feb 04 2022
%F A002295 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^11). - _Seiichi Manyama_, Jun 16 2025
%e A002295 There are a(2)=6 sextic trees (vertex degree <= 6 and 6 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 6 trees yields 6*6 + binomial(6,2) = 51 = a(3) such trees.
%p A002295 A002295:=n->binomial(6*n, n)/(5*n + 1); seq(A002295(n), n=0..20); # _Wesley Ivan Hurt_, Jan 29 2014
%p A002295 n:=20:G:=series(RootOf(g = 1+x*g^6, g),x=0,n+1):seq(coeff(G,x,k),k=0..n); # _Robert FERREOL_, Apr 03 2015
%t A002295 Table[Binomial[6n, n]/(5n + 1), {n, 0, 20}] (* _Stefan Steinerberger_, Apr 06 2006 *)
%o A002295 (Magma) [Binomial(6*n, n)/(5*n + 1): n in [0..20]]; // _Vincenzo Librandi_, Mar 17 2015
%o A002295 (PARI) A002295(n)=binomial(6*n,n)/(5*n+1) \\ _M. F. Hasler_, Apr 08 2015
%o A002295 (GAP) List([0..22],n->Binomial(6*n,n)/(5*n+1)); # _Muniru A Asiru_, Nov 01 2018
%Y A002295 Cf. A001764, A002293, A002294, A002296, A130564.
%Y A002295 Fifth column of triangle A062993.
%K A002295 easy,nonn,nice
%O A002295 0,3
%A A002295 _N. J. A. Sloane_
%E A002295 More terms from _Stefan Steinerberger_, Apr 06 2006
%E A002295 Edited by _M. F. Hasler_, Apr 08 2015