cp's OEIS Frontend

A002294 a(n) = binomial(5*n, n)/(4*n + 1).

%I A002294 M3977 N1646 #228 Jun 16 2025 08:47:05
%S A002294 1,1,5,35,285,2530,23751,231880,2330445,23950355,250543370,2658968130,
%T A002294 28558343775,309831575760,3390416787880,37377257159280,
%U A002294 414741863546285,4628362722856425,51912988256282175,584909606696793885,6617078646960613370
%N A002294 a(n) = binomial(5*n, n)/(4*n + 1).
%C A002294 From _Wolfdieter Lang_, Sep 14 2007: (Start)
%C A002294 a(n), n >= 1, enumerates quintic trees (rooted, ordered, incomplete) with n vertices (including the root).
%C A002294 This is the Pfaff-Fuss-Catalan sequence C^{m}_n for m = 5. See the Graham et al. reference, p. 347. eq. 7.66. See also the Pólya-Szegő reference.
%C A002294 Also 5-Raney sequence. See the Graham et al. reference, pp. 346-347. (End)
%C A002294 a(n) = A258708(3*n, 2*n) for n > 0. - _Reinhard Zumkeller_, Jun 23 2015
%C A002294 Conjecturally, a(n) is the number of 4-uniform words on the alphabet [n] that avoid the patterns 231 and 221 (see the Defant and Kravitz link). - _Colin Defant_, Sep 26 2018
%C A002294 From Stillwell (1995), p. 62: "Eisenstein's Theorem. If y^5 + y = x, then y has a power series expansion y = x - x^5 + 10*x^9/2^1 - 15 * 14 * x^13/3! + 20 * 19 * 18*x^17/4! - ...." - _Michael Somos_, Sep 19 2019
%C A002294 a(n) is the total number of down steps before the first up step in all 4_1-Dyck paths of length 5*n. A 4_1-Dyck path is a lattice path with steps (1, 4), (1, -1) that starts and ends at y = 0 and stays above the line y = -1. - _Sarah Selkirk_, May 10 2020
%C A002294 Dropping the first 1 (starting from 1, 5, 35, ... with offset 1), the series reversion gives 1, -5, 15, -35, 70, ... (again offset 1), essentially A000332 and row 5 of A027555. - _R. J. Mathar_, Aug 17 2023
%C A002294 Number of rooted polyominoes composed of n hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. A stereographic projection of the {6,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - _Robert A. Russell_, Jan 27 2024
%C A002294 This is instance k = 5 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564. - Wolfdieter Lang, Feb 05 2024
%D A002294 Archiv der Mathematik u. Physik, 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 Lame, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathematiques pures et appliquees, publie par Joseph Liouville. T. III. IV.", Archiv der Mathematik u. Physik, 1 (1841), pp. 193ff; see especially p. 198.
%D A002294 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 23.
%D A002294 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
%D A002294 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 A002294 Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nürnberg, Jul 27 1994.
%D A002294 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002294 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002294 T. D. Noe, <a href="/A002294/b002294.txt">Table of n, a(n) for n = 0..100</a>
%H A002294 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 A002294 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp. 337-338.
%H A002294 Joerg Arndt, <a href="http://arxiv.org/abs/1405.6503">Subset-lex: did we miss an order?</a>, arXiv:1405.6503 [math.CO], 2014-2015.
%H A002294 A. Asinowski, B. Hackl, and S. Selkirk, <a href="https://arxiv.org/abs/2007.15562">Down step statistics in generalized Dyck paths</a>, arXiv:2007.15562 [math.CO], 2020.
%H A002294 Paul Barry, <a href="https://arxiv.org/abs/2001.08799">Characterizations of the Borel triangle and Borel polynomials</a>, arXiv:2001.08799 [math.CO], 2020.
%H A002294 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 A002294 L. W. Beineke and R. E. Pippert, <a href="https://doi.org/10.1017/S0017089500002305">On the enumeration of planar trees of hexagons</a>, Glasgow Math. J., 15 (1974), 131-147.
%H A002294 L. W. Beineke and R. E. Pippert, <a href="/A004127/a004127.pdf">On the enumeration of planar trees of hexagons</a>, Glasgow Math. J., 15 (1974), 131-147. [Annotated scanned copy]
%H A002294 Frits Beukers, <a href="http://www.ams.org/notices/201401/">Hypergeometric functions, how special are they?</a>, Notices Amer. Math. Soc. 61(1) (2014), 48-56. MR3137256
%H A002294 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 A002294 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%H A002294 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 A002294 C. Defant and N. Kravitz, <a href="https://arxiv.org/abs/1809.09158">Stack-sorting for words</a>, arXiv:1809.09158 [math.CO], 2018.
%H A002294 R. W. Gosper, <a href="http://gosper.org/newsrope.pdf">Rope around the earth</a>
%H A002294 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 A002294 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 A002294 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 A002294 Forrest M. Hilton, <a href="https://arxiv.org/abs/2408.01353">Finite Dynamical Laminations</a>, arXiv:2408.01353 [math.DS], 2024. See p. 7.
%H A002294 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 A002294 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=287">Encyclopedia of Combinatorial Structures 287</a>.
%H A002294 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 A002294 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 A002294 Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, <a href="http://arxiv.org/abs/1511.00080">Pattern Avoidance in Task-Precedence Posets</a>, arXiv:1511.00080 [math.CO], 2015-2016.
%H A002294 Karol A. Penson and Karol Zyczkowski, <a href="http://dx.doi.org/10.1103/PhysRevE.83.061118">Product of Ginibre matrices : Fuss-Catalan and Raney distribution</a>, <a href="http://arxiv.org/abs/1103.3453/">arXiv version</a>, arXiv:1103.3453 [math-ph], 2011.
%H A002294 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 A002294 John Stillwell, <a href="https://doi.org/10.1007/BF03024901">Eisenstein's footnote</a>, Math. Intelligencer 17(2) (1995), 58-62. MR1336074 (96d:01024)
%H A002294 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 A002294 L. 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).
%H A002294 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan_number">Fuss-Catalan number</a>.
%H A002294 N. J. Wildberger and Dean Rubine, <a href="https://doi.org/10.1080/00029890.2025.2460966">A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode</a>, Amer. Math. Monthly (2025). See section 12.
%H A002294 S. Yakoubov, <a href="http://arxiv.org/abs/1310.2979">Pattern Avoidance in Extensions of Comb-Like Posets</a>, arXiv:1310.2979 [math.CO], 2013-2014.
%H A002294 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 A002294 For the connection with the solution of the quintic, hypergeometric series, and Lagrange inversion, see Beukers (2014). - _N. J. A. Sloane_, Mar 12 2014
%F A002294 G.f.: hypergeometric([1, 2, 3, 4] / 5, [2, 3, 5] / 4, x * 5^5 / 4^4). - _Michael Somos_, Mar 17 2011
%F A002294 O.g.f. A(x) satisfies A(x) = 1 + x * A(x)^5 = 1 / (1 - x * A(x)^4).
%F A002294 Given g.f. A(x) then z = t * A(t^4) satisfies 0 = z^5 - z + t. - _Michael Somos_, Mar 17 2011
%F A002294 a(n) = binomial(5*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 A002294 a(n) = upper left term in M^n, M = the production matrix:
%F A002294    1,  1;
%F A002294    4,  4,  1;
%F A002294   10, 10,  4,  1;
%F A002294   20, 20, 10,  4,  1;
%F A002294   ...
%F A002294 where (1, 4, 10, 20, ...) is the tetrahedral sequence, A000292. - _Gary W. Adamson_, Jul 08 2011
%F A002294 D-finite with recurrence: 8*n*(4*n+1)*(2*n-1)*(4*n-1)*a(n) - 5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1) = 0. - _R. J. Mathar_, Dec 02 2014
%F A002294 a(n) = binomial(5*n + 1, n)/(5*n + 1) = A062993(n+3,3). - _Robert FERREOL_, Apr 03 2015
%F A002294 a(0) = 1; a(n) = Sum_{i1 + i2 + ... + i5 = n - 1} a(i1) * a(i2) * ... *a(i5) for n >= 1. - _Robert FERREOL_, Apr 03 2015
%F A002294 From _Ilya Gutkovskiy_, Jan 15 2017: (Start)
%F A002294 O.g.f.: 5F4([1/5, 2/5, 3/5, 4/5, 1]; [1/2, 3/4, 1, 5/4]; 3125*x/256).[Cancellation of the 1s, see G.f. the above. - _Wolfdieter Lang_, Feb 05 2024]
%F A002294 E.g.f.: 4F4([1/5, 2/5, 3/5, 4/5]; [1/2, 3/4, 1, 5/4]; 3125*x/256).
%F A002294 a(n) ~ 5^(5*n + 1/2)/(sqrt(Pi) * 2^(8*n + 7/2) * n^(3/2)). (End)
%F A002294 x*A'(x)/A(x) = (A(x) - 1)/(- 4*A(x) + 5) = x + 9*x^2 + 91*x^3 + 969*x^4 + ... is the o.g.f. of A163456. Cf. A001764 and A002293 - A002296. - _Peter Bala_, Feb 04 2022
%F A002294 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^9). - _Seiichi Manyama_, Jun 16 2025
%e A002294 There are a(2) = 5 quintic trees (vertex degree <= 5 and 5 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these five trees yields 5*5 + binomial(5,2) = 35 = a(3) such trees.
%e A002294 G.f. = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 + 231880*x^7 + ...
%e A002294 G.f. = t + t^5 + 5*t^9 + 35*t^13 + 285*t^17 + 2530*t^21 + 23751*t^25 + 231880*t^29 + ...
%p A002294 seq(binomial(5*k+1,k)/(5*k+1),k=0..30); # _Robert FERREOL_, Apr 03 2015
%p A002294 n:=30:G:=series(RootOf(g = 1+x*g^5, g),x=0,n+1):seq(coeff(G,x,k),k=0..n); # _Robert FERREOL_, Apr 03 2015
%t A002294 CoefficientList[InverseSeries[ Series[ y - y^5, {y, 0, 100}], x], x][[Range[2, 100, 4]]]
%t A002294 Table[Binomial[5n,n]/(4n+1),{n,0,20}] (* _Harvey P. Dale_, Dec 30 2011 *)
%t A002294 a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1, 2, 3, 4}/5, {2, 3, 5}/4, x 5^5/4^4], {x, 0, n}]; (* _Michael Somos_, May 06 2015 *)
%t A002294 a[ n_] := With[{m = 4 n + 1}, SeriesCoefficient[ InverseSeries @ Series[ x - x^5, {x, 0, m}], {x, 0, m}]]; (* _Michael Somos_, May 06 2015 *)
%o A002294 (PARI) {a(n) = binomial( 5 * n, n) / (4*n + 1)}; /* _Michael Somos_, Mar 17 2011 */
%o A002294 (PARI) {a(n) = if( n<0, 0, n = 4*n + 1; polcoeff( serreverse( x - x^5 + x * O(x^n) ), n))}; /* _Michael Somos_, Mar 17 2011 */
%o A002294 (Magma) [ Binomial(5*n,n)/(4*n+1): n in [0..100]]; // _Vincenzo Librandi_, Mar 24 2011
%o A002294 (Haskell)
%o A002294 a002294 n = a002294_list !! n
%o A002294 a002294_list = [a258708 (3 * n) (2 * n) | n <- [1..]]
%o A002294 -- _Reinhard Zumkeller_, Jun 23 2015
%o A002294 (GAP) List([0..22],n->Binomial(5*n,n)/(4*n+1)); # _Muniru A Asiru_, Nov 01 2018
%Y A002294 Cf. A001764, A002296, A258708, A346647 (binomial transform), A346665 (inverse binomial transform).
%Y A002294 Fourth column of triangle A062993.
%Y A002294 Polyominoes: A221184{n-1} (oriented), A004127 (unoriented), A369473 (chiral), A143546 (achiral), A002293 {5,oo}, A002295 {7,oo}.
%Y A002294 Cf. A130564.
%K A002294 easy,nonn,nice
%O A002294 0,3
%A A002294 _N. J. A. Sloane_
%E A002294 More terms from _Olivier Gérard_, Jul 05 2001