A002294 -id:A002294 - cp's OEIS frontend

A365178 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x).

1, 1, 5, 30, 210, 1595, 12791, 106574, 913562, 8004861, 71375653, 645536234, 5907683486, 54605672300, 509043322720, 4780441915832, 45182744331388, 429472919087158, 4102806757542542, 39370967793387086, 379335734835510622, 3668220243145708341

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..988

Cf. A002294, A365180, A365181, A365182, A365183.
Cf. A364475, A365184.
Cf. A002293, A364987.

a(n) = sum(k=0, n, binomial(k, n-k)*binomial(4*k, k)/(3*k+1));

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(k,n-k)/(4*k+1) = Sum_{k=0..n} binomial(k,n-k) * A002293(k).

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

A305143 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x)^4 dx ) / A(x), for n > 0.

Original entry on oeis.org

1, 1, 13, 316, 10667, 447576, 22094626, 1242995118, 78081518451, 5400194995057, 406998451178896, 33165909456647704, 2904055577822356346, 271843880829531635092, 27087966494039897011884, 2862718283883222686998584, 319838550858171357010036323, 37670084296166551957561304631

Offset: 0

Paul D. Hanna, May 31 2018

nonn

It is remarkable that this sequence should consist entirely of integers.

Note: 0 = [x^n] exp( n * Integral F(x)^4 dx ) / F(x) holds for n > 0 when F(x) = 1 + x*F(x)^5 is a g.f. of A002294.

			O.g.f.: A(x) = 1 + x + 13*x^2 + 316*x^3 + 10667*x^4 + 447576*x^5 + 22094626*x^6 + 1242995118*x^7 + 78081518451*x^8 + 5400194995057*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^2*Integral A(x)^4 dx)/A(x) begins:
  n=0: [1, -1, -12, -291, -9904, -419430, -20878908, ...];
  n=1: [1, 0, -21/2, -284, -78945/8, -420765, -336068285/16, ...];
  n=2: [1, 3, 0, -235, -9540, -421722, -21319776, ...];
  n=3: [1, 8, 75/2, 0, -61985/8, -408126, -345111453/16, ...];
  n=4: [1, 15, 132, 861, 0, -328662, -20947980, ...];
  n=5: [1, 24, 651/2, 3384, 226143/8, 0, -268775133/16, ...];
  n=6: [1, 35, 672, 9621, 117020, 1187142, 0, ...];
  n=7: [1, 48, 2475/2, 23180, 2891295/8, 5049117, 960763011/16, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2*Integral A(x)^4 dx)/A(x), for n > 0.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 27*x^2 + 658*x^3 + 22135*x^4 + 924702*x^5 + 45461602*x^6 + 2548558008*x^7 + 159620140335*x^8 + ...
A(x)^3 = 1 + 3*x + 42*x^2 + 1027*x^3 + 34443*x^4 + 1432833*x^5 + 70159774*x^6 + 3919323204*x^7 + 244746587643*x^8 + ...
A(x)^4 = 1 + 4*x + 58*x^2 + 1424*x^3 + 47631*x^4 + 1973476*x^5 + 96250266*x^6 + 5358025992*x^7 + 333596305267*x^8 + ...
exp( Integral A(x)^4 dx) = 1 + x + 9*x^2/2! + 373*x^3/3! + 35809*x^4/4! + 5918961*x^5/5! + 1461206521*x^6/6! + 496585571749*x^7/7! + 220438988917953*x^8/8! + ...
A'(x)/A(x) = 1 + 25*x + 910*x^2 + 41117*x^3 + 2166366*x^4 + 128865058*x^5 + 8487954042*x^6 + 611163126189*x^7 + 47668752953875*x^8 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..250

Cf. A305137, A305138, A305139, A305140, A305142.

{a(n) = my(A=[1],m); for(i=1,n+1, m=#A; A=concat(A,0); A[m+1] = Vec( exp(m^2*intformal(Ser(A)^4)) / Ser(A) )[m+1] );A[n+1]}
for(n=0,20,print1(a(n),", "))

(a(n)/n!)^(1/n) tends to 25/4. - Vaclav Kotesovec, Oct 19 2020

A386812 a(n) = Sum_{k=0..n} binomial(5*n+1,k).

Original entry on oeis.org

1, 7, 67, 697, 7547, 83682, 942649, 10739176, 123388763, 1427090845, 16593192942, 193774331494, 2271115189673, 26700463884244, 314735943548632, 3718522618187472, 44021808206431579, 522080025971331983, 6201449551502245321, 73767447652621434695, 878599223738760686422

Offset: 0

Seiichi Manyama, Aug 03 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A000302, A160906, A386811.
Cf. A002294, A371739.
Cf. A001449, A079589, A079678, A371753, A385632.

[&+[Binomial(5*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 21 2025

Table[Sum[Binomial[5*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 21 2025 *)

a(n) = sum(k=0, n, binomial(5*n+1, k));

a(n) = [x^n] 1/((1-2*x) * (1-x)^(4*n+1)).
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+k,k).
D-finite with recurrence 8*n*(2754528070303487*n -4672004545621835)*(4*n-3)*(2*n-1) *(4*n-1)*a(n) +(-5828620079131711179*n^5 -135826272187971586019*n^4 +779361612339655552281*n^3 -1570139520911413863589*n^2 +1419656431480813021170*n -487668485184225269400)*a(n-1) +40*(-21123668262204329085*n^5 +243394620512022153401*n^4 -982249084763267479011*n^3 +1849334401749026834935*n^2 -1662134287466221884960*n +573649997457991096080)*a(n-2) +6400*(5*n-13)*(5*n-11)*(2475036532470005*n-2376524337096748)*(5*n-9)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Aug 03 2025
a(n) = 2^(5*n+1) - binomial(5*n+1, n)*(hypergeom([1, -1-4*n], [1+n], -1) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k). - Seiichi Manyama, Aug 07 2025
G.f.: g^2/((2-g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 12 2025
From Seiichi Manyama, Aug 16 2025: (Start)
G.f.: 1/(1 - x*g^3*(10-3*g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: B(x)^2/(1 + 3*(B(x)-1)/5), where B(x) is the g.f. of A001449. (End)
a(n) ~ 5^(5*n + 3/2) / (3*sqrt(Pi*n) * 2^(8*n + 3/2)). - Vaclav Kotesovec, Aug 21 2025

A004127 Number of planar hexagon trees with n hexagons.

Original entry on oeis.org

1, 1, 3, 12, 68, 483, 3946, 34485, 315810, 2984570, 28907970, 285601251, 2868869733, 29227904840, 301430074416, 3141985563575, 33059739636198, 350763452126835, 3749420616902637, 40348040718155170, 436827335493148600

Offset: 1

N. J. A. Sloane

nonn

Number of nonequivalent dissections of a polygon into n hexagons by nonintersecting diagonals up to rotation and reflection. - Andrew Howroyd, Nov 20 2017

Number of unoriented polyominoes composed of n hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. For unoriented polyominoes, chiral pairs are counted as one. - Robert A. Russell, Jan 23 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..925
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
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]
Index entries for sequences related to trees

Column k=6 of A295260.
Cf. A002294.
Polyominoes: A221184{n-1} (oriented), A369473 (chiral), A143546 (achiral), A005040 {5,oo}, A005419 {7,oo}.

T := proc(n) if floor(n)=n then binomial(5*n+1,n)/(5*n+1) else 0 fi end: U := proc(n) if n mod 2 = 0 then binomial(5*n/2+1, n/2)/(5*n/2+1) else 6*binomial((5*n+1)/2,(n-1)/2)/(5*n+1) fi end: S := n->T(n)/4/(2*n+1)+T(n/2)/6+(5*n-2)*T((n-1)/3)/6/(2*n+1)+T((n-1)/6)/6+7*U(n)/12: seq(S(n),n=1..25); (Emeric Deutsch)

p=6; Table[(Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) + If[OddQ[n], If[OddQ[p], Binomial[(p-1)n/2, (n-1)/2]/n, (p+1)Binomial[((p-1)n-1)/2, (n-1)/2]/((p-2)n+2)], 3Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+Plus @@ Map[EulerPhi[ # ]Binomial[((p-1)n+1)/#, (n-1)/# ]/((p-1)n+1)&, Complement[Divisors[GCD[p, n-1]], {1, 2}]])/2, {n, 1, 20}] (* Robert A. Russell, Dec 11 2004 *)

See Theorem 3 on p. 142 in the Beineke-Pippert paper; also the Maple and Mathematica codes here.
a(n) ~ 5^(5*n + 1/2) / (sqrt(Pi) * n^(5/2) * 2^(8*n + 13/2)). - Vaclav Kotesovec, Mar 13 2016
a(n) = A221184(n-1) - A369473(n) = (A221184(n-1) + A143546(n)) / 2 = A369473(n) + A143546(n). - Robert A. Russell, Jan 23 2024

More terms from Emeric Deutsch, Jan 22 2004

A118969 a(n) = 2binomial(5n+1,n)/(4*n+2).

Original entry on oeis.org

1, 2, 11, 80, 665, 5980, 56637, 556512, 5620485, 57985070, 608462470, 6474009360, 69682358811, 757366074080, 8300675584120, 91634565938880, 1018002755977245, 11372548404732930, 127677890035721025, 1439777493407492640

Offset: 0

Paul Barry, May 07 2006

A quadrisection of A118968.

If y = x + 2*x^3 + x^5, the series reversion is x = y - 2*y^3 + 11*y^5 - 80*y^7 + 665*y^9 - ... - R. J. Mathar, Sep 29 2012

			a(3) = 80 = sum of top row terms in M^n = (35 + 35 + 9 + 1).

Vincenzo Librandi, Table of n, a(n) for n = 0..100
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.
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).
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

Cf. A000292, A118968, A006013, A233832, A234505, A234868, A002294.

[2*Binomial(5*n+1,n)/(4*n+2): n in [0..20]]; // Vincenzo Librandi, Aug 12 2011

Table[2*Binomial[5n+1,n]/(4n+2),{n,0,20}] (* Harvey P. Dale, Aug 21 2011 *)

a(n)=2*binomial(5*n+1,n)/(4*n+2); \\ Joerg Arndt, Apr 20 2013

From Gary W. Adamson, Aug 11 2011: (Start)
a(n) is sum of top row terms in M^n, where M is an infinite square production matrix with the tetrahedral series in each column (A000292), as follows:
   1,  1,  0,  0,  0,  0, ...
   4,  1,  1,  0,  0,  0, ...
  10, 10,  4,  1,  0,  0, ...
  20, 20, 10,  4,  1,  0, ...
  35, 35, 20, 10,  4,  1, ...
  ... (End)
G.f.: hypergeom([1/5, 2/5, 3/5, 4/5],[1/2, 3/4, 5/4], 3125*x/256)^2. - Mark van Hoeij, Apr 19 2013
a(n) = 2*binomial(5n+1,n-1)/n for n>0, a(0)=1. - Bruno Berselli, Jan 19 2014
D-finite with recurrence 8*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n) - 5*(5*n+1)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1) = 0. - R. J. Mathar, Oct 10 2014
G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(x)^2)^2. - Ilya Gutkovskiy, Nov 13 2021

A365183 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x*A(x)^4).

Original entry on oeis.org

1, 1, 5, 34, 268, 2299, 20838, 196326, 1903524, 18868861, 190356231, 1948055058, 20173907384, 211020478270, 2226243632838, 23660868061422, 253099278807684, 2722819049879436, 29439894433161189, 319749417998303470, 3486914150183526920

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..939
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 13.

Cf. A002294, A365178, A365180, A365181, A365182.
Cf. A006605, A255673, A365189.
Cf. A364989.

a(n) = sum(k=0, n\2, binomial(n-k, k)*binomial(4*n+1, n-k))/(4*n+1);

a(n) = (1/(4*n+1)) * Sum_{k=0..floor(n/2)} binomial(n-k,k) * binomial(4*n+1,n-k).

A233668 a(n) = 6binomial(5n + 6,n)/(5*n + 6).

Original entry on oeis.org

1, 6, 45, 380, 3450, 32886, 324632, 3290040, 34034715, 357919100, 3815041230, 41124015036, 447534498320, 4910258796240, 54257308779600, 603260892430960, 6744185681876505, 75764901779438850, 854867886710698755, 9683529727259434200

Offset: 0

Tim Fulford, Dec 14 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 5, r = 6.

C. H. Pah, M. R. Wahiddin, Combinatorial Interpretation of Raney Numbers and Tree Enumerations, Open Journal of Discrete Mathematics, 2015, 5, 1-9; http://www.scirp.org/journal/ojdm; http://dx.doi.org/10.4236/ojdm.2015.51001

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A002294, A118969, A143546, A118971, A233669, A233736, A233737, A233738.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

[6*Binomial(5*n+6,n)/(5*n+6): n in [0..30]];

Table[6 Binomial[5 n + 6, n]/(5 n + 6), {n, 0, 30}]

PARI
```
a(n) = 6*binomial(5*n+6,n)/(5*n+6);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/6))^6+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, here p = 5, r = 6.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^6), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/6) is the o.g.f. for A002294. (End)
D-finite with recurrence 8*n*(4*n+5)*(2*n+3)*(4*n+3)*a(n) -5*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A334785 a(n) is the total number of down steps before the first up step in all 3_2-Dyck paths of length 4*n. A 3_2-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -2.

Original entry on oeis.org

0, 3, 13, 74, 480, 3363, 24794, 189540, 1488744, 11941820, 97412601, 805602850, 6738919408, 56918898330, 484750343700, 4158094853640, 35891774969112, 311529010178628, 2717299393716836, 23806014817182600, 209389427777770240, 1848322153489496355

Offset: 0

Sarah Selkirk, May 11 2020

			For n = 1, there are the 3_2-Dyck paths UDDD, DUDD, DDUD. Before the first up step there are a(1) = 0 + 1 + 2 = 3 down steps in total.

Stefano Spezia, Table of n, a(n) for n = 0..1000
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A001764, A002293, A002294, A334786, A334787.

a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n+1) - Binomial[4*n+2, n]/(n+1); Array[a, 22, 0]

a(0) = 0 and a(n) = 3*binomial(4*n, n)/(n+1) - binomial(4*n+2, n)/(n+1) for n > 0.

a(n) ~ c*2^(8*n)*3^(-3*n)/n^(3/2), where c = (11/9)*sqrt(2/(3*Pi)). - Stefano Spezia, Oct 19 2022

A349290 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^4)).

Original entry on oeis.org

1, 2, 11, 96, 1001, 11456, 139013, 1756596, 22867421, 304560171, 4130200726, 56836946342, 791689962811, 11140615233281, 158140107648676, 2261708608884896, 32559326010349817, 471428798399646336, 6860801662510005266, 100302910051255600486

Offset: 0

Ilya Gutkovskiy, Nov 13 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002294, A007317, A199475, A346647, A349289, A349291, A349292, A349293.

nmax = 19; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^4)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 3 k, 4 k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 19}]

a(n) = sum(k=0, n, binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1)); \\ Michel Marcus, Nov 14 2021

a(n) = Sum_{k=0..n} binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1).
a(n) = F([1/5, 2/5, 3/5, 4/5, (1+n)/3, (2+n)/3, (3+n)/3, -n], [1/4, 1/2, 1/2, 3/4, 3/4, 1, 5/4], -3^3*5^5/2^16), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 13 2021
a(n) ~ sqrt(1 + 3*r) / (2 * 5^(3/4) * sqrt(2*Pi*(1-r)) * n^(3/2) * r^(n + 1/4)), where r = 0.0631152861998150860738633360987635931... is the root of the equation 5^5 * r = 4^4 * (1-r)^4. - Vaclav Kotesovec, Nov 14 2021
a(n) = 1 + Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} a(i) * a(j) * a(k) * a(l) * a(m). - Seiichi Manyama, Jul 10 2025

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A365178 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x).

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

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A305143 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x)^4 dx ) / A(x), for n > 0.

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A386812 a(n) = Sum_{k=0..n} binomial(5*n+1,k).

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Magma

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A004127 Number of planar hexagon trees with n hexagons.

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A118969 a(n) = 2*binomial(5*n+1,n)/(4*n+2).

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A365183 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x*A(x)^4).

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A365178 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x).

A118969 a(n) = 2binomial(5n+1,n)/(4*n+2).

A365183 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x*A(x)^4).

A233668 a(n) = 6binomial(5n + 6,n)/(5*n + 6).