A002294 a(n) = binomial(5*n, n)/(4*n + 1). (Formerly M3977 N1646)

1, 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, 2658968130, 28558343775, 309831575760, 3390416787880, 37377257159280, 414741863546285, 4628362722856425, 51912988256282175, 584909606696793885, 6617078646960613370

Offset

0,3

Comments

From Wolfdieter Lang, Sep 14 2007: (Start)

a(n), n >= 1, enumerates quintic trees (rooted, ordered, incomplete) with n vertices (including the root).

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.

Also 5-Raney sequence. See the Graham et al. reference, pp. 346-347. (End)

a(n) = A258708(3*n, 2*n) for n > 0. - Reinhard Zumkeller, Jun 23 2015

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

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

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

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

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

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

References

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.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 23.

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).

Links

T. D. Noe, Table of n, a(n) for n = 0..100

V. E. Adler and A. B. Shabat, Volterra chain and Catalan numbers, arXiv:1810.13198 [nlin.SI], 2018.

Joerg Arndt, Matters Computational (The Fxtbook), pp. 337-338.

Joerg Arndt, Subset-lex: did we miss an order?, arXiv:1405.6503 [math.CO], 2014-2015.

A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.

Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.

L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147.

L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147. [Annotated scanned copy]

Frits Beukers, Hypergeometric functions, how special are they?, Notices Amer. Math. Soc. 61(1) (2014), 48-56. MR3137256

Wun-Seng Chou, Tian-Xiao He, and Peter J.-S. Shiue, On the Primality of the Generalized Fuss-Catalan Numbers, J. Int. Seqs., Vol. 21 (2018), #18.2.1.

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

C. Defant and N. Kravitz, Stack-sorting for words, arXiv:1809.09158 [math.CO], 2018.

R. W. Gosper, Rope around the earth

F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.

F. Harary, E. M. Palmer, and R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974.

Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.

V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 287.

R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, preprint, 1980.

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.

Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.

Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, arXiv version, arXiv:1103.3453 [math-ph], 2011.

John Stillwell, Eisenstein's footnote, Math. Intelligencer 17(2) (1995), 58-62. MR1336074 (96d:01024)

B. Sury, Generalized Catalan numbers: linear recursion and divisibility, JIS 12 (2009), Article 09.7.5.

L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).

Wikipedia, Fuss-Catalan number.

S. Yakoubov, Pattern Avoidance in Extensions of Comb-Like Posets, arXiv:1310.2979 [math.CO], 2013-2014.

Formula

For the connection with the solution of the quintic, hypergeometric series, and Lagrange inversion, see Beukers (2014). - N. J. A. Sloane, Mar 12 2014

G.f.: hypergeometric([1, 2, 3, 4] / 5, [2, 3, 5] / 4, x * 5^5 / 4^4). - Michael Somos, Mar 17 2011

O.g.f. A(x) satisfies A(x) = 1 + x * A(x)^5 = 1 / (1 - x * A(x)^4).

Given g.f. A(x) then z = t * A(t^4) satisfies 0 = z^5 - z + t. - Michael Somos, Mar 17 2011

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.

a(n) = upper left term in M^n, M = the production matrix:

1, 1;

4, 4, 1;

10, 10, 4, 1;

20, 20, 10, 4, 1;

...

where (1, 4, 10, 20, ...) is the tetrahedral sequence, A000292. - Gary W. Adamson, Jul 08 2011

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

a(n) = binomial(5*n + 1, n)/(5*n + 1) = A062993(n+3,3). - Robert FERREOL, Apr 03 2015

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

From Ilya Gutkovskiy, Jan 15 2017: (Start)

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]

E.g.f.: 4F4([1/5, 2/5, 3/5, 4/5]; [1/2, 3/4, 1, 5/4]; 3125*x/256).

a(n) ~ 5^(5*n + 1/2)/(sqrt(Pi) * 2^(8*n + 7/2) * n^(3/2)). (End)

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

Example

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.
G.f. = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 + 231880*x^7 + ...
G.f. = t + t^5 + 5*t^9 + 35*t^13 + 285*t^17 + 2530*t^21 + 23751*t^25 + 231880*t^29 + ...

Maple

seq(binomial(5*k+1, k)/(5*k+1), k=0..30); # Robert FERREOL, Apr 03 2015
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

Mathematica

CoefficientList[InverseSeries[ Series[ y - y^5, {y, 0, 100}], x], x][[Range[2, 100, 4]]]
Table[Binomial[5n, n]/(4n+1), {n, 0, 20}] (* Harvey P. Dale, Dec 30 2011 *)
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 *)
a[ n_] := With[{m = 4 n + 1}, SeriesCoefficient[ InverseSeries @ Series[ x - x^5, {x, 0, m}], {x, 0, m}]]; (* Michael Somos, May 06 2015 *)

Prog

(PARI) {a(n) = binomial( 5 * n, n) / (4*n + 1)}; /* Michael Somos, Mar 17 2011 */
(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 */
(Magma) [ Binomial(5*n, n)/(4*n+1): n in [0..100]]. // Vincenzo Librandi, Mar 24 2011
(Haskell)
a002294 n = a002294_list !! n
a002294_list = [a258708 (3 * n) (2 * n) | n <- [1..]]
-- Reinhard Zumkeller, Jun 23 2015
(GAP) List([0..22], n->Binomial(5*n, n)/(4*n+1)); # Muniru A Asiru, Nov 01 2018

Crossrefs

Cf. A001764, A002296, A258708, A346647 (binomial transform), A346665 (inverse binomial transform).

Fourth column of triangle A062993.

Polyominoes: A221184{n-1} (oriented), A004127 (unoriented), A369473 (chiral), A143546 (achiral), A002293 {5,oo}, A002295 {7,oo}.

Cf. A130564.

Sequence in context: A322666 A248053 A369157 * A369128 A051406 A000356

Adjacent sequences: A002291 A002292 A002293 * A002295 A002296 A002297

Keyword

easy,nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Olivier Gérard, Jul 05 2001

Status

approved