A163456 -id:A163456 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

			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 + ...

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).

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.
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 25.
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.
Forrest M. Hilton, Finite Dynamical Laminations, arXiv:2408.01353 [math.DS], 2024. See p. 7.
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.
Yuan (Friedrich) Qiu, Joe Sawada, and Aaron Williams, Maximize the Rightmost Digit: Gray Codes for Restricted Growth Strings, WALCOM 2025. See p. 5.
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.
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.
S. Yakoubov, Pattern Avoidance in Extensions of Comb-Like Posets, arXiv:1310.2979 [math.CO], 2013-2014.
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

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.

List([0..22],n->Binomial(5*n,n)/(4*n+1)); # Muniru A Asiru, Nov 01 2018

a002294 n = a002294_list !! n
a002294_list = [a258708 (3 * n) (2 * n) | n <- [1..]]
-- Reinhard Zumkeller, Jun 23 2015

[ Binomial(5*n,n)/(4*n+1): n in [0..100]]; // Vincenzo Librandi, Mar 24 2011

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

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 *)

{a(n) = binomial( 5 * n, n) / (4*n + 1)}; /* Michael Somos, Mar 17 2011 */

{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 */

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
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^9). - Seiichi Manyama, Jun 16 2025

More terms from Olivier Gérard, Jul 05 2001

A069271 a(n) = binomial(4n+1,n)2/(3*n+2).

Original entry on oeis.org

1, 2, 9, 52, 340, 2394, 17710, 135720, 1068012, 8579560, 70068713, 580034052, 4855986044, 41043559340, 349756577100, 3001701610320, 25921837477692, 225083787458904, 1963988670706228, 17211860478150800, 151433425446423120

Offset: 0

Henry Bottomley, Mar 12 2002

nonn

This sequence counts the set B_n of plane trees defined in the Poulalhon and Schaeffer link (Definition 2.2 and Section 4.2, Proposition 4). - David Callan, Aug 20 2014
a(n) is the number of lattice paths of length 4n starting and ending on the x-axis consisting of steps {(1, 1), (1, -3)} that remain on or above the line y=-1. - Sarah Selkirk, Mar 31 2020
a(n) is the number of ordered pairs of 4-ary trees with a (summed) total of n internal nodes. - Sarah Selkirk, Mar 31 2020

			a(3) = C(4*3+1,3)*2/(3*3+2) = C(13,3)*2/11 = 286*2/11 = 52.
a(3) = 52 since the top row of M^3 = (22, 22, 7, 1).
1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 + 135720*x^7 + ...
q + 2*q^3 + 9*q^5 + 52*q^7 + 340*q^9 + 2394*q^11 + 17710*q^13 + 135720*q^15 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Thomas Anderson, T. Bruce McLean, Homeira Pajoohesh, and Chasen Smith , The combinatorics of all regular flexagons, Eu. J. Combinat. 31 (2010) 72-80, Theorem 2.
Roland Bacher, Fair Triangulations, arXiv:0710.0960 [math.CO], 2007.
Esther Banaian, Elise Catania, Christian Gaetz, Miranda Moore, Gregg Musiker, and Kayla Wright, Twists, Higher Dimer Covers, and Web Duality for Grassmannian Cluster Algebras, arXiv:2507.15211 [math.CO], 2025. See p. 28.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Gi-Sang Cheon, S.-T. Jin and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137.
Emmanuel Guitter, The distance-dependent two-point function of triangulations: a new derivation from old results, Ann. Inst. Henri Poincaré Comb. Phys. Interact. Vol. 4 (2017), 177-211. DOI: 10.4171/AIHPD/38.
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.
Ionut E. Iacob, T. Bruce McLean and Hua Wang, The V-flex, Triangle Orientation, and Catalan Numbers in Hexaflexagons, The College Mathematics Journal, Vol. 43, No. 1 (January 2012), pp. 6-10.
Jean-Christophe Novelli and Jean-Yves Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.
C. O. Oakley and R. J. Wisner, Flexagons, Am. Math. Monthly 64 (3) (1957) 143-154, u_{3k+2}.
Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, arXiv:1103.3453 [math-ph], 2011.
Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, Phys. Rev. E 83, 061118, 2011.
Dominique Poulalhon and Gilles Schaeffer, Optimal Coding and Sampling of Triangulations, in Automata, Languages and Programming, Lecture Notes in Computer Science, Volume 2719, 2003, pp 1080-1094.
Sarah J. Selkirk, On a generalisation of k-Dyck paths, MSc Thesis, 2019.

Cf. A002293, A006013, A006632, A069270 for similar generalized Catalan sequences.

Cf. A000260, A115141, A163456, A224274.

[2*Binomial(4*n+1, n)/(3*n+2): n in [0..20]];  // Bruno Berselli, Mar 04 2011

BB:=[T,{T=Prod(Z,Z,Z,F,F),F=Sequence(B),B=Prod(F,F,F,Z)}, unlabeled]: seq(count(BB,size=i),i=3..23); # Zerinvary Lajos, Apr 22 2007

f[n_] := 2 Binomial[4 n + 1, n]/(3 n + 2); Array[f, 21, 0] (* Robert G. Wilson v *)

a(n)=if(n<0,0,polcoeff(serreverse(x/(1+x^2)^2+O(x^(2*n+2))),2*n+1)) /* Ralf Stephan */

{a(n) =  binomial(4*n + 2, n)*2 / (2*n + 1)} /* Michael Somos, Mar 28 2012 */

{a(n) =  local(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = (1 + x * A^2)^2); polcoeff( A, n))} /* Michael Somos, Mar 28 2012 */

a(n) = A069270(n+1, n) = A005810(n)*A016813(n)/A060544(n+1)
O.g.f. A(x) satisfies 2*x^2*A(x)^3 = 1-2*x*A(x)-sqrt(1-4*x*A(x)). - Vladimir Kruchinin, Feb 23 2011
a(n) is the sum of top row terms in M^n, where M is the infinite square production matrix with the triangular series in each column as follows, with the rest zeros:
   1,  1,  0, 0, 0, 0, ...
   3,  3,  1, 0, 0, 0, ...
   6,  6,  3, 1, 0, 0, ...
  10, 10,  6, 3, 1, 0, ...
  15, 15, 10, 6, 3, 1, ...
  ... - Gary W. Adamson, Aug 11 2011
Given g.f. A(x) then B(x) = x * A(x^2) satisfies x = B(x) / (1 + B(x)^2)^2. - Michael Somos, Mar 28 2012
Given g.f. A(x) then A(x) = (1 + x * A(x)^2)^2. - Michael Somos, Mar 28 2012
a(n) / (n+1) = A000260(n). - Michael Somos, Mar 28 2012
REVERT transform is A115141. - Michael Somos, Mar 28 2012
D-finite with recurrence 3*n*(3*n+2)*(3*n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Jun 07 2013
a(n) = 2*binomial(4n+1,n-1)/n for n>0, a(0)=1. - Bruno Berselli, Jan 19 2014
G.f.: hypergeom([1/2, 3/4, 5/4], [4/3, 5/3], (256/27)*x). - Robert Israel, Aug 24 2014
From Peter Bala, Oct 08 2015: (Start)
O.g.f. A(x) = (1/x) * series reversion (x/C(x)^2), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. Cf. A163456.
(1/2)*x*A'(x)/A(x) is the o.g.f. for A224274. (End)
E.g.f.: hypergeom([1/2, 3/4, 5/4], [1, 4/3, 5/3], (256/27)*x). - Karol A. Penson, Jun 26 2017
a(n) = binomial(4*n+2,n)/(2*n+1). - Alexander Burstein, Nov 08 2021

A224274 a(n) = binomial(4*n,n)/4.

Original entry on oeis.org

1, 7, 55, 455, 3876, 33649, 296010, 2629575, 23535820, 211915132, 1917334783, 17417133617, 158753389900, 1451182990950, 13298522298180, 122131734269895, 1123787895356412, 10358022441395860, 95615237915961100, 883829035553043580, 8179808679272664720, 75788358475481302185

Offset: 1

Gary Detlefs, Apr 02 2013

In general, binomial(k*n,n)/k = binomial(k*n-1,n-1).
Sequences in the OEIS related to this identity are:
. C(2n,n) = A000984, C(2n,n)/2 = A001700;
. C(3n,n) = A005809, C(3n,n)/3 = A025174;
. C(4n,n) = A005810, C(4n,n)/4 = a(n);
. C(5n,n) = A001449, C(5n,n)/5 = A163456;
. C(6n,n) = A004355, C(6n,n)/6 is not in the OEIS.
Conjecture: a(n) == 1 (mod n^3) iff n is an odd prime.
It is known that a(p) == 1(mod p^3) for prime p >= 3. See Mestrovic, Section 3. - Peter Bala, Oct 09 2015

			For n=2, binomial(4*n,n) = binomial(8,2) = 8*7/2 = 28, so a(2) = 28/4 = 7. - _Michael B. Porter_, Jul 12 2016

G. C. Greubel, Table of n, a(n) for n = 1..1000
Dmitry Kruchinin and Vladimir Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.
Romeo Meštrović, Lucas’ theorem: its generalizations, extensions and applications (1878-2014), arXiv:1409.3820v1 [math.NT], 2014.
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.

Cf. A001700, A001764, A025174, A163456, A002293, A069271, A004331, A005810, A052203, A257633, A262977, A227726.

[Binomial(4*n,n) div 4: n in [1..25]]; // Vincenzo Librandi, Jun 03 2015

Maple
```
seq(binomial(4*n,n)/4, n=1..17);
```

Table[Binomial[4 n, n]/4, {n, 30}] (* Vincenzo Librandi, Jun 03 2015 *)

a(n) = binomial(4*n,n)/4; /* Joerg Arndt, Apr 02 2013 */

a(n) = binomial(4*n,n)/4 = A005810(n)/4.
a(n) = binomial(4*n-1,n-1).
G.f.: A(x) = B'(x)/B(x), where B(x) = 1 + x*B(x)^4 is g.f. of A002293. - Vladimir Kruchinin, Aug 13 2015
From Peter Bala, Oct 08 2015: (Start)
a(n) = 1/2*[x^n] (C(x)^2)^n, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. Cf. A163456.
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 22*x^3 + ... is the o.g.f. for A002293.
exp( 2*Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*x + 9*x^2 + 52*x^3 + ... is the o.g.f. for A069271. (End)
From Peter Bala, Nov 04 2015: (Start)
With an offset of 1, the o.g.f. equals f(x)*g(x)^3, where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A262977 (k = -1), A005810 (k = 0), A052203 (k = 1), A257633 (k = 2) and A004331 (k = 4). (End)
a(n) = 1/5*[x^n] (1 + x)/(1 - x)^(3*n + 1) = 1/5*[x^n]( 1/C(-x) )^(5*n), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. Cf. A227726. - Peter Bala, Jul 12 2016
a(n) ~ 2^(8*n-3/2)*3^(-3*n-1/2)*n^(-1/2)/sqrt(Pi). - Ilya Gutkovskiy, Jul 12 2016
O.g.f.: A(x) = f(x)/(1 - 3*f(x)), where f(x) = series reversion (x/(1 + x)^4) = x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + ... is the o.g.f. of A002293 with the initial term omitted. Cf. A025174. - Peter Bala, Feb 03 2022
Right-hand side of the identities (1/3)*Sum_{k = 0..n} (-1)^(n+k)*C(x*n,n-k)*C((x+3)*n+k-1,k) = C(4*n,n)/4 and (1/4)*Sum_{k = 0..n} (-1)^k*C(x*n,n-k)*C((x-4)*n+k-1,k) = C(4*n,n)/4, both valid for n >= 1 and x arbitrary. - Peter Bala, Feb 28 2022
Right-hand side of the identity (1/3)*Sum_{k = 0..2*n} (-1)^k*binomial(5*n-k-1,2*n-k)*binomial(3*n+k-1,k) = binomial(4*n,n)/4. - Peter Bala, Mar 09 2022
a(n) = [x^n] G(x)^n, where G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ... is the g.f. of A001764. - Peter Bala, Oct 17 2024

A118970 a(n) = 3*binomial(5n+2,n)/(4n+3).

Original entry on oeis.org

1, 3, 18, 136, 1155, 10530, 100688, 996336, 10116873, 104819165, 1103722620, 11777187240, 127067830773, 1383914371728, 15194457001440, 167996704221280, 1868870731122405, 20903064321375315, 234927317665726686

Offset: 0

Paul Barry, May 07 2006

A quadrisection of A118968.

Convolved with A118969 (1, 2, 11, 80, 665, ...) = A002294: (1, 5, 35, 285, 2530, ...) - Gary W. Adamson, Nov 07 2011

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B33.

Michael De Vlieger, Table of n, a(n) for n = 0..924
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.
Henri Muehle and Philippe Nadeau, A Poset Structure on the Alternating Group Generated by 3-Cycles, arXiv:1803.00540 [math.CO], 2018.

Cf. A118969, A002294, A000108, A163456.

ogf := series(RootOf(A = 1 + x * A^5,A)^3, x=0, 30); # Mark van Hoeij, Apr 22 2013

Array[3 Binomial[5 # + 2, #]/(4 # + 3) &, 19, 0] (* Michael De Vlieger, May 30 2018 *)
CoefficientList[Series[HypergeometricPFQ[{3/5,4/5,6/5,7/5},{1,5/4,3/2,7/4},(5^5/4^4)x],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Oct 01 2024 *)

a(n)=3*binomial(5*n+2,n)/(4*n+3); \\ Joerg Arndt, Apr 23 2013

G.f.: F^3 where F is the g.f. of A002294. - Mark van Hoeij, Apr 23 2013
8*n*(4*n+1)*(2*n+1)*(4*n+3)*a(n) -5*(5*n+1)*(5*n+2)*(5*n-2)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Dec 02 2014
From Peter Bala, Oct 08 2015: (Start)
O.g.f. A(x) = (1/x) * series reversion ( x/C(x)^3 ), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108.
(1/3)*x*A'(x)/A(x) = x + 9*x^2 + 91*x^3 + 969*x^4 + ... is the o.g.f. for A163456. (End)
E.g.f.: hypergeom([3/5, 4/5, 6/5, 7/5], [1, 5/4, 3/2, 7/4], (5^5/4^4)*x). - Stefano Spezia, Oct 01 2024

A260687 Triangular array with n-th row giving coefficients of polynomial Product_{k = 2..n} (k + n*t) for n >= 1.

Original entry on oeis.org

1, 2, 2, 6, 15, 9, 24, 104, 144, 64, 120, 770, 1775, 1750, 625, 720, 6264, 20880, 33480, 25920, 7776, 5040, 56196, 250096, 571095, 708295, 453789, 117649, 40320, 554112, 3127040, 9433088, 16486400, 16744448, 9175040, 2097152, 362880, 5973264, 41229324, 156498804

Offset: 1

Peter Bala, Nov 16 2015

Related to A220883 and A251592.

			Triangle begins
...1
...2      2
...6     15       9
..24    104     144      64
.120    770    1775    1750     625
.720   6264   20880   33480   25920    7776
5040  56196  250096  571095  708295  453789  117649
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998

P. Bala, Fractional iteration of a series inversion operator

A000142 (column 0), A000169 (main diagonal), A006675 (column 1). Cf. A001700, A025174, A056856, A163456, A220883, A224274, A251592.

seq(seq(coeff(mul(n*t + k, k = 2 .. n), t, i), i = 0..n-1), n = 1..10);

E.g.f. (with constant term 1 included): A(x,t) = [ 1/x*Revert( x*(1 - x)^t ) ]^(1/t) = Sum_{n >= 0} 1/(n*t + 1)*binomial(n*t + n,n)*x^n = 1 + x + (2 + 2*t)*x^2/2! + (2 + 3*t)*(3 + 3*t)*x^3/3! + (2 + 4*t)*(3 + 4*t)*(4 + 4*t)*x^4/4! + ..., where Revert denotes the series reversion operator with respect to x.
In the notation of the Bala link, A(x,t) = I^t(1/(1 - x)) where I^t is a fractional inversion operator.
A(x,t) = B_(1+t)(x), where B_t(x) is the e.g.f. for A251592 and is the generalized binomial series of Lambert. See Graham et al., Section 5.4 and Section 7.5.
A(x,t)^m = Sum_{n >= 0} m/(n*t + m)*binomial(n*t + n + m - 1,n)*x^n = 1 + m*x + m*(2*t + m + 1)*x^2/2! + m*(3*t + m + 1)*(3*t + m + 2)*x^3/3! + m*(4*t + m + 1)*(4*t + m + 2)*(4*t + m + 3)*x^4/4! + ....
A(x,t)^t = 1 + t*x + t(1 + 3*t)*x^2/2! + t*(1 + 4*t)*(2 + 4*t)*x^3/3! + t*(1 + 5*t)*(2 + 5*t)*(3 + 5*t)*x^4/4! + ... is the e.g.f for A220883 with an extra constant term 1 and an extra factor of t included.
t*log( A(x,t) ) = t*x + t*(1 + 2*t)*x^2/2! + t*(1 + 3*t)*(2 + 3*t)*x^3/3! + t*(1 + 4*t)*(2 + 4*t)*(3 + 4*t)*x^4/4! + ... is the e.g.f for A056856.
For n = 1,2,3,..., the sequence [x^n] A(x,t)^n = [1, (2*t + 3), (3*t + 4)*(3*t + 5)/2!, (4*t + 5)*(4*t + 6)*(4*t + 7)/3!, ...]. This sequence has the following specializations:
t = 0: [1, 3, 10, 35, 126, ...] = A001700 (with different offset).
t = 1: [1, 5, 28, 165, 1001, ...] = A025174.
t = 2: [1, 7, 55, 455, 3876, ...] = A224274.
t = 3: [1, 9, 91, 969, 10626, ...] = A163456.

cp's OEIS Frontend

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

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A069271 a(n) = binomial(4*n+1,n)*2/(3*n+2).

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A224274 a(n) = binomial(4*n,n)/4.

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A118970 a(n) = 3*binomial(5n+2,n)/(4n+3).

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A260687 Triangular array with n-th row giving coefficients of polynomial Product_{k = 2..n} (k + n*t) for n >= 1.

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A318593 Number of n-member subsets of [5*n] whose elements sum to a multiple of five.

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

A069271 a(n) = binomial(4n+1,n)2/(3*n+2).