A143547 -id:A143547 - cp's OEIS frontend

A143554 G.f. A(x) satisfies A(x) = 1 + xA(x)^5A(-x)^4.

1, 1, 1, 5, 9, 55, 117, 775, 1785, 12350, 29799, 211876, 527085, 3818430, 9706503, 71282640, 184138713, 1366368375, 3573805950, 26735839650, 70625252863, 531838637759, 1416298046436, 10723307329700, 28748759731965, 218658647805780, 589546754316126

Offset: 0

Paul D. Hanna, Aug 24 2008

nonn

Number of achiral noncrossing partitions composed of n blocks of size 9. - Andrew Howroyd, Feb 08 2024

			G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 9*x^4 + 55*x^5 + 117*x^6 + 775*x^7 +...
Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then
G(x^2) = A(x)*A(-x) and A(x) = G(x^2) + x*G(x^2)^5 where
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
G(x)^5 = 1 + 5*x + 55*x^2 + 775*x^3 + 12350*x^4 + 211876*x^5 +...

Andrew Howroyd, Table of n, a(n) for n = 0..500
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1. - From _N. J. A. Sloane_, Jul 12 2011

Column k=9 of A369929 and k=10 of A370062.
Cf. A143338, A143546, A143547, A143550, A062994 (bisection).
Cf. A143551, A143552, A143553.
Cf. A143047.

terms = 25;
A[] = 1; Do[A[x] = 1 + x A[x]^5 A[-x]^4 + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 24 2018 *)

{a(n)=my(A=1+x*O(x^n));for(i=0,n,A=1+x*A^5*subst(A^4,x,-x));polcoef(A,n)}

{a(n)=my(m=n\2,p=4*(n%2)+1);binomial(9*m+p-1,m)*p/(8*m+p)}

G.f. satisfies: A(x) = [A(x)*A(-x)] + x*[A(x)*A(-x)]^5.
G.f. satisfies: A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.
a(2n) = binomial(9*n,n)/(8*n+1); a(2n+1) = binomial(9*n+4,n)*5/(8*n+5).
a(0) = 1; a(n) = Sum_{i, j, k, l, m>=0 and i+2*j+2*k+2*l+2*m=n-1} a(i) * a(2*j) * a(2*k) * a(2*l) * a(2*m). - Seiichi Manyama, Jul 07 2025
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_9>=0 and x_1+x_2+...+x_9=n-1} (-1)^(x_1+x_2+x_3+x_4) * Product_{k=1..9} a(x_k). - Seiichi Manyama, Jul 09 2025

A233835 a(n) = 8binomial(7n + 8, n)/(7*n + 8).

Original entry on oeis.org

1, 8, 84, 1008, 13090, 179088, 2542512, 37106784, 553270671, 8391423040, 129058047580, 2008018827360, 31550226597162, 499892684834368, 7978140653296800, 128138773298754240, 2069603881026760323, 33593111381834512200, 547698081896206040800, 8965330544164089648000, 147285313888568167177866

Offset: 0

Tim Fulford, Dec 16 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 7, r = 8.

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, A002296, A233832, A233833, A143547, A233834, A233835, A233907, A233908.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

[8*Binomial(7*n+8, n)/(7*n+8): n in [0..30]];

Table[8 Binomial[7 n + 8, n]/(7 n + 8), {n, 0, 30}]

PARI
```
a(n) = 8*binomial(7*n+8,n)/(7*n+8);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/8))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 7, r = 8.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^8), 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)^(1/8) is the o.g.f. for A002296. (End)

A233834 a(n) = 5binomial(7n+5,n)/(7*n+5).

Original entry on oeis.org

1, 5, 45, 500, 6200, 82251, 1142295, 16398200, 241379325, 3623534200, 55262073757, 853814730600, 13335836817420, 210225027967325, 3340362288091500, 53443628421286320, 860246972339613855, 13921016318025200505, 226352372251889455000, 3696160728052814340000

Offset: 0

Tim Fulford, Dec 16 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p = 7, r = 5.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, 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, A002296, A233832, A233833, A143547, A130565, A233835, A233907, A233908, A002296, A069271, A118970, A212073, A234465, A234510, A234571, A235339.

[5*Binomial(7*n+5, n)/(7*n+5): n in [0..30]];

Table[5 Binomial[7 n + 5, n]/(7 n + 5), {n, 0, 30}]

PARI
```
a(n) = 5*binomial(7*n+5,n)/(7*n+5);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/5))^5+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 7, r = 5.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^5), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/5) is the o.g.f. for A002296. - Peter Bala, Oct 14 2015

A369929 Array read by antidiagonals: T(n,k) is the number of achiral noncrossing partitions composed of n blocks of size k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 3, 6, 1, 1, 1, 1, 3, 5, 7, 10, 1, 1, 1, 1, 4, 5, 16, 12, 20, 1, 1, 1, 1, 4, 7, 18, 31, 30, 35, 1, 1, 1, 1, 5, 7, 31, 35, 102, 55, 70, 1, 1, 1, 1, 5, 9, 34, 64, 136, 213, 143, 126, 1

Offset: 0

Andrew Howroyd, Feb 07 2024

T(n,2*k-1) is the number of achiral noncrossing k-gonal cacti with n polygons.

			Array begins:
===============================================
n\k| 1  2   3   4    5    6    7    8     9 ...
---+-------------------------------------------
0  | 1  1   1   1    1    1    1    1     1 ...
1  | 1  1   1   1    1    1    1    1     1 ...
2  | 1  1   1   1    1    1    1    1     1 ...
3  | 1  2   2   3    3    4    4    5     5 ...
4  | 1  3   3   5    5    7    7    9     9 ...
5  | 1  6   7  16   18   31   34   51    55 ...
6  | 1 10  12  31   35   64   70  109   117 ...
7  | 1 20  30 102  136  296  368  651   775 ...
8  | 1 35  55 213  285  663  819 1513  1785 ...
9  | 1 70 143 712 1155 3142 4495 9304 12350 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1.
Wikipedia, Fuss-Catalan number.
Wikipedia, Noncrossing partition.

Columns are: A000012, A001405(n-1), A047749 (k=3), A369930 (k=4), A143546 (k=5), A143547 (k=7), A143554 (k=9), A192893 (k=11).

Cf. A070914, A303694, A303929, A361236, A361239, A370060, A370062.

\\ u(n,k,r) are Fuss-Catalan numbers.
u(n,k,r) = {r*binomial(k*n + r, n)/(k*n + r)}
e(n,k) = {sum(j=0, n\2, u(j, k, 1+(n-2*j)*k/2))}
T(n, k)={if(n==0, 1, if(k%2, if(n%2, 2*u(n\2, k, (k+1)/2), u(n/2, k, 1) + u(n/2-1, k, k)), e(n, k) + if(n%2, u(n\2, k, k/2)))/2)}

T(n,k) = 2*A303929(n,k) - A303694(n,k).

T(n,2*k-1) = 2*A361239(n,k) - A361236(n,k).

A370062 Array read by antidiagonals: T(n,k) is the number of achiral dissections of a polygon into n k-gons by nonintersecting diagonals, n >= 1, k >= 3.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 4, 7, 5, 1, 1, 3, 5, 9, 12, 5, 1, 1, 4, 6, 18, 22, 30, 14, 1, 1, 4, 7, 21, 35, 52, 55, 14, 1, 1, 5, 8, 34, 51, 136, 140, 143, 42, 1, 1, 5, 9, 38, 70, 190, 285, 340, 273, 42, 1, 1, 6, 10, 55, 92, 368, 506, 1155, 969, 728, 132

Offset: 1

Andrew Howroyd, Feb 08 2024

The polygon prior to dissection will have n*(k-2)+2 sides.

			Array begins:
=============================================
n\k|  3   4   5    6    7    8    9    10 ...
---+-----------------------------------------
1  |  1   1   1    1    1    1    1     1 ...
2  |  1   1   1    1    1    1    1     1 ...
3  |  1   2   2    3    3    4    4     5 ...
4  |  2   3   4    5    6    7    8     9 ...
5  |  2   7   9   18   21   34   38    55 ...
6  |  5  12  22   35   51   70   92   117 ...
7  |  5  30  52  136  190  368  468   775 ...
8  | 14  55 140  285  506  819 1240  1785 ...
9  | 14 143 340 1155 1950 4495 6545 12350 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
Wikipedia, Fuss-Catalan number

Columns are A208355(n-1), A047749 (k=4), A369472 (k=5), A143546 (k=6), A143547 (k=8), A143554 (k=10), A192893 (k=12).

Cf. A070914 (rooted), A295224 (oriented), A295260 (unoriented), A369929, A370060 (achiral rooted at cell).

\\ here u is Fuss-Catalan sequence with p = k-1.
u(n, k, r) = {r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
T(n, k) = {(if(n%2, u((n-1)/2, k, k\2), if(k%2, u(n/2-1, k, k-1), u(n/2, k, 1))))}
for(n=1, 9, for(k=3, 10, print1(T(n, k), ", ")); print);

T(n,k) = 2*A295260(n,k) - A295224(n,k).
T(n,2*k+1) = A370060(n,2*k+1).
T(n,2*k) = A369929(n,2*k-1).

A233832 a(n) = 2binomial(7n+2,n)/(7*n+2).

Original entry on oeis.org

1, 2, 15, 154, 1827, 23562, 320866, 4540200, 66096459, 983592304, 14894775896, 228784720710, 3555866673450, 55819631671902, 883738853546472, 14094715154157680, 226245021605612955, 3652242142988400570, 59254515909624764575, 965678197027521177200

Offset: 0

Tim Fulford, Dec 16 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=2.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
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.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number
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. A000108, A002296, A233833 - A233835, A143547, A130565, A233907, A233908.

[2*Binomial(7*n+2, n)/(7*n+2): n in [0..30]];

Table[2 Binomial[7 n + 2, n]/(7 n + 2), {n, 0, 30}]

PARI
```
a(n) = 2*binomial(7*n+2,n)/(7*n+2);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/2))^2+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=2.
a(n) = 2*binomial(7n+1, n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 6F6(2/7,3/7,4/7,5/7,6/7,8/7; 1/2,2/3,5/6,1,7/6,4/3; 823543*x/46656).
a(n) ~ 7^(7*n+3/2)/(sqrt(Pi)*3^(6*n+5/2)*4^(3*n+1)*n^(3/2)). (End)

A143553 G.f. A(x) satisfies A(x) = 1 + xA(x)^5A(-x)^3.

Original entry on oeis.org

1, 1, 2, 14, 50, 432, 1818, 17082, 77714, 763967, 3637718, 36786268, 180481258, 1860798032, 9324573430, 97502825964, 496344066386, 5245970686152, 27032002846992, 288124627083382, 1499144278319270, 16087838913122064

Offset: 0

Paul D. Hanna, Aug 24 2008

nonn

			G.f. A(x) = 1 + x + 2*x^2 + 14*x^3 + 50*x^4 + 432*x^5 + 1818*x^6 +...
Related expansions:
A(x)^5 = 1 + 5*x + 20*x^2 + 120*x^3 + 635*x^4 + 4301*x^5 + 25360*x^6 +...
A(-x)^3 = 1 - 3*x + 9*x^2 - 55*x^3 + 252*x^4 - 1818*x^5 + 9560*x^6 -+...
A(x)*A(-x) = 1 + 3*x^2 + 76*x^4 + 2776*x^6 + 118940*x^8 +...
[A(x)*A(-x)]^8 = 1 + 24*x^2 + 860*x^4 + 36488*x^6 + 1700198*x^8 +...

Cf. A143551, A143552, A143554.
Cf. A143046, A143547.
Cf. A143338, A143550.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^5*subst(A^3,x,-x));polcoeff(A,n)}

G.f. satisfies: A(x) + A(-x) = 1 + [A(x)*A(-x)] + x^2*[A(x)*A(-x)]^8.

a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_8>=0 and x_1+x_2+...+x_8=n-1} (-1)^(x_1+x_2+x_3) * Product_{k=1..8} a(x_k). - Seiichi Manyama, Jul 08 2025

A233833 a(n) = 3binomial(7n+3, n)/(7*n+3).

Original entry on oeis.org

1, 3, 24, 253, 3045, 39627, 543004, 7718340, 112752783, 1682460520, 25533901536, 392912889915, 6116090678334, 96133810101609, 1523687678528400, 24324750346691480, 390786855500604195, 6313161418594235271, 102494297789621214400, 1671366110239940499000

Offset: 0

Tim Fulford, Dec 16 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=3.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
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.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A002296, A233832, A143547, A233834, A130565, A233835, A233907, A233908.

[3*Binomial(7*n+3, n)/(7*n+3): n in [0..30]];

Table[3 Binomial[7 n + 3, n]/(7 n + 3), {n, 0, 30}]

PARI
```
a(n)=3*binomial(7*n+3,n)/(7*n+3);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/3))^3+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=3.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 6F6(3/7,4/7,5/7,6/7,8/7,9/7; 2/3,5/6,1,7/6,4/3,3/2; 823543*x/46656).
a(n) ~ 7^(7*n+5/2)/(sqrt(Pi)*3^(6*n+5/2)*4^(3*n+2)*n^(3/2)). (End)

A233907 9binomial(7n+9, n)/(7*n+9).

Original entry on oeis.org

1, 9, 99, 1218, 16065, 222138, 3178140, 46656324, 698868216, 10639125640, 164128169205, 2560224004884, 40314178429707, 639948824981928, 10230035192533800, 164541833894991240, 2660919275605834701, 43239781879996449825, 705687913212419321800, 11561996402992103418000, 190100812111989146008641

Offset: 0

Tim Fulford, Dec 17 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=9.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, 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.

Cf. A000108, A002296, A233832, A233833, A143547, A233834, A130565, A233835, A233908.

[9*Binomial(7*n+9, n)/(7*n+9): n in [0..30]];

Table[9 Binomial[7 n + 9, n]/(7 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(7*n+9,n)/(7*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=9.

D-finite with recurrence 72*n*(6*n+5)*(3*n+2)*(2*n+3)*(3*n+4)*(6*n+7)*a(n) -7*(7*n+4)*(7*n+8)*(7*n+5)*(7*n+2)*(7*n+6)*(7*n+3)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A233908 10binomial(7n+10,n)/(7*n+10).

Original entry on oeis.org

1, 10, 115, 1450, 19425, 271502, 3915100, 57821940, 870238200, 13298907050, 205811513765, 3218995093860, 50802419972395, 808016193159000, 12938696992921000, 208419656266988904, 3374960506795660365, 54907659530154222000, 897060906625956765000

Offset: 0

Tim Fulford, Dec 17 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=10.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, 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, A002296, A233832 - A233835, A143547, A130565, A233907.

[10*Binomial(7*n+10, n)/(7*n+10): n in [0..30]]; // Vincenzo Librandi, Dec 23 2013

Table[10 Binomial[7 n + 10, n]/(7 n + 10), {n, 0, 40}] (* Vincenzo Librandi, Dec 23 2013 *)

PARI
```
a(n) = 10*binomial(7*n+10,n)/(7*n+10);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/10))^10+x*O(x^n)); polcoeff(B, n)}

72*n*(6*n+5)*(3*n+5)*(2*n+3)*(3*n+4)*(6*n+7)*a(n) -7*(7*n+4)*(7*n+8)*(7*n+5)*(7*n+9)*(7*n+6)*(7*n+3)*a(n-1)=0. - R. J. Mathar, Dec 22 2013

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=10.

cp's OEIS Frontend

A143554 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5*A(-x)^4.

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A233835 a(n) = 8*binomial(7*n + 8, n)/(7*n + 8).

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A233834 a(n) = 5*binomial(7*n+5,n)/(7*n+5).

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A369929 Array read by antidiagonals: T(n,k) is the number of achiral noncrossing partitions composed of n blocks of size k.

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A370062 Array read by antidiagonals: T(n,k) is the number of achiral dissections of a polygon into n k-gons by nonintersecting diagonals, n >= 1, k >= 3.

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A233832 a(n) = 2*binomial(7*n+2,n)/(7*n+2).

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A143553 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5*A(-x)^3.

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A143554 G.f. A(x) satisfies A(x) = 1 + xA(x)^5A(-x)^4.

A233835 a(n) = 8binomial(7n + 8, n)/(7*n + 8).

A233834 a(n) = 5binomial(7n+5,n)/(7*n+5).

A233832 a(n) = 2binomial(7n+2,n)/(7*n+2).

A143553 G.f. A(x) satisfies A(x) = 1 + xA(x)^5A(-x)^3.

A233833 a(n) = 3binomial(7n+3, n)/(7*n+3).

A233907 9binomial(7n+9, n)/(7*n+9).

A233908 10binomial(7n+10,n)/(7*n+10).