A143546 -id:A143546 - cp's OEIS frontend

A233669 a(n) = 7binomial(5n+7, n)/(5*n+7).

1, 7, 56, 490, 4550, 44051, 439824, 4496388, 46834095, 495260150, 5303177880, 57385471962, 626548297648, 6893781417320, 76362138282400, 850867975145160, 9530515916642385, 107249427630005661, 1211964598880990640, 13747501038498835300

Offset: 0

Tim Fulford, Dec 14 2013

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

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, A002294, A118969, A143546, A118971, A233668, A233736, A233737, A233738.

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

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

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

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

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=5, r=7.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 4F4(7/5,8/5,9/5,11/5; 1,9/4,5/2,11/4; 3125*x/256).
a(n) ~ 7*5^(5*n+13/2)/(sqrt(Pi)*2^(8*n+31/2)*n^(3/2)). (End)

A233736 a(n) = 8binomial(5n + 8, n)/(5*n + 8).

Original entry on oeis.org

1, 8, 68, 616, 5850, 57536, 581196, 5995184, 62891499, 668922800, 7197169980, 78195588168, 856708896784, 9454328800896, 104997940138300, 1172624772468960, 13161188646791865, 148375147999406328, 1679436658449372744, 19078164706488179600

Offset: 0

Tim Fulford, Dec 15 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=5, r=8.

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, A002294, A118969, A143546, A118971, A233668, A233669, A233737, A233738.

[8*Binomial(5*n+8,n)/(5*n+8): n in [0..30]]; // Vincenzo Librandi, Dec 16 2013

Table[8 Binomial[5 n + 8, n]/(5 n + 8), {n, 0, 40}] (* Vincenzo Librandi, Dec 16 2013 *)

PARI
```
a(n) = 8*binomial(5*n+8,n)/(5*n+8);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/8))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=5, r=8.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 5F5(8/5,9/5,2,11/5,12/5; 1,9/4,5/2,11/4,3; 3125*x/256).
a(n) ~ 5^(5*n+15/2)/(sqrt(Pi)*2^(8*n+29/2)*n^(3/2)). (End)

A233737 a(n) = 9binomial(5n+9, n)/(5*n+9).

Original entry on oeis.org

1, 9, 81, 759, 7371, 73656, 752913, 7838298, 82832706, 886322710, 9583986555, 104568156819, 1149793519368, 12728471356944, 141747219186705, 1586867219265060, 17848735288114995, 201607141031660871, 2285899896222757346, 26008027474874327190, 296840444852078282610, 3397721117411729991960

Offset: 0

Tim Fulford, Dec 15 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=5, 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, A002294, A118969, A143546, A118971, A233668, A233669, A233736, A233738.

[9*Binomial(5*n+9,n)/(5*n+9): n in [0..30]];

Table[9 Binomial[5 n + 9, n]/(5 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(5*n+9,n)/(5*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=5, r=9.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 5F5(9/5,2,11/5,12/5,13/5; 1,5/2,11/4,3,13/4; 3125*x/256).
a(n) ~ 9*5^(5*n+17/2)/(sqrt(Pi)*2^(8*n+39/2)*n^(3/2)). (End)

A143338 G.f. A(x) satisfies A(x) = 1 + xA(x)^3A(-x).

Original entry on oeis.org

1, 1, 2, 8, 26, 127, 478, 2536, 10250, 56900, 239880, 1370272, 5940054, 34607146, 153018932, 904441648, 4058644842, 24254529036, 110096276440, 663665021280, 3040205250984, 18455364854839, 85176971647470, 520059936017128

Offset: 0

Paul D. Hanna, Aug 09 2008

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 26*x^4 + 127*x^5 + 478*x^6 +...
Compare bisections of A(x)^2, A(x)^2*A(-x), and A(x)^4*A(-x)^2:
A(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 72*x^4 + 338*x^5 + 1378*x^6 + 6952*x^7 +...
A(x)^2*A(-x) = 1 + x + 5*x^2 + 11*x^3 + 72*x^4 + 191*x^5 + 1378*x^6 + 3979*x^7 +...
A(x)^4*A(-x)^2 = 1 + 2*x + 11*x^2 + 32*x^3 + 191*x^4 + 636*x^5 + 3979*x^6 +...
Related expansions:
A(x)^3 = 1 + 3*x + 9*x^2 + 37*x^3 + 144*x^4 + 669*x^5 + 2882*x^6 + 14229*x^7 +...
A(x)^3*A(-x) = 1 + 2*x + 8*x^2 + 26*x^3 + 127*x^4 + 478*x^5 + 2536*x^6 +...
A(x)^3*A(-x)^2 = 1 + x + 8*x^2 + 14*x^3 + 127*x^4 + 264*x^5 + 2536*x^6 +...

Cf. A047749, A143549, A143551.

Cf. A143546.

{a(n)=local(A=1+x+O(x^21));for(i=0,n,A=1+x*A^3*subst(A,x,-x));polcoeff(A,n)}

a(0) = 1; a(n) = Sum_{i, j, k, l>=0 and i+j+k+l=n-1} (-1)^i * a(i) * a(j) * a(k) * a(l). - Seiichi Manyama, Jul 08 2025

A192893 Number of symmetric 11-ary factorizations of the n-cycle (1,2...n).

Original entry on oeis.org

1, 1, 1, 6, 11, 81, 176, 1406, 3311, 27636, 68211, 585162, 1489488, 13019909, 33870540, 300138696, 793542167, 7105216833, 19022318084, 171717015470, 464333035881, 4219267597578, 11502251937176, 105085831400550, 288417894029200, 2647012241261856, 7306488667126803

Offset: 0

N. J. A. Sloane, Jul 12 2011

nonn

The six sequences displayed in Table 1 of the Bousquet-Lamathe reference are A047749, A143546, A143547, A143554, this sequence, and A192894. From this one should be able to guess a g.f.

Number of achiral noncrossing partitions composed of n blocks of size 11. - Andrew Howroyd, Feb 08 2024

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.

Column k=11 of A369929 and k=12 of A370062.

Cf. A143048.

a(n)={my(m=n\2, p=5*(n%2)+1); binomial(11*m+p-1, m)*p/(10*m+p)} \\ Andrew Howroyd, Feb 08 2024

From Andrew Howroyd, Feb 08 2024: (Start)
a(2n) = binomial(11*n,n)/(10*n+1); a(2n+1) = binomial(11*n+5,n)*6/(10*n+6).
G.f. A(x) satisfies A(x) = 1 + x*A(x)^6*A(-x)^5. (End)
From Seiichi Manyama, Jul 07 2025: (Start)
G.f. A(x) satisfies A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2), where G(x) = 1 + x*G(x)^11 is the g.f. of A230388.
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_6>=0 and x_1+2*(x_2+x_3+...+x_6)=n-1} a(x_1) * Product_{k=2..6} a(2*x_k). (End)
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_11>=0 and x_1+x_2+...+x_11=n-1} (-1)^(x_1+x_2+x_3+x_4+x_5) * Product_{k=1..11} a(x_k). - Seiichi Manyama, Jul 09 2025

a(11) onwards from Andrew Howroyd, Jan 26 2024

a(0)=1 prepended by Andrew Howroyd, Feb 08 2024

A369473 Number of chiral pairs of polyominoes composed of n hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}.

Original entry on oeis.org

7, 50, 448, 3810, 34200, 314655, 2982040, 28897440, 285577500, 2868769045, 29227672960, 301429078080, 3141983233130, 33059729519325, 350763428176480, 3749420512083472, 40348040467611800, 436827334389425980

Offset: 4

Robert A. Russell, Jan 23 2024

A stereographic projection of the {6,oo} tiling on the Poincaré disk can be obtained via the Christensson link. Each member of a chiral pair is a reflection but not a rotation of the other.

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

Polyominoes: A221184(n-1) (oriented), A004127 (unoriented), A143546 (achiral), A369471 {5,oo}.

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)-Binomial[((p-1)n+1)/2, (n-1)/2]/((p-1)n+1)], Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+DivisorSum[GCD[p, n-1], EulerPhi[#]Binomial[((p-1)n+1)/#, (n-1)/#]/((p-1)n+1)&, #>1&])/2, {n, 4, 30}]

a(n) = A221184(n-1) - A004127(n) = (A221184(n-1) - A143546(n)) / 2 = A004127(n) - A143546(n).

A192894 Number of symmetric 13-ary factorizations of the n-cycle (1,2...n).

Original entry on oeis.org

1, 1, 1, 7, 13, 112, 247, 2310, 5525, 53998, 135408, 1360289, 3518515, 36017352, 95223414, 988172368, 2655417765, 27844071255, 75769712590, 801012669457, 2201663313200, 23428926096576, 64924369564353, 694644371065372, 1938034271677595, 20829931845958872, 58448142042957576

Offset: 0

N. J. A. Sloane, Jul 12 2011

nonn

The six sequences displayed in Table 1 of the Bousquet-Lamathe reference are A047749, A143546, A143547, A143554, A192893, A192894. From this one should be able to guess a g.f.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1.

Column k=13 of A369929 and k=14 of A370062.

Cf. A143049.

From Seiichi Manyama, Jul 07 2025: (Start)
G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x)*A(-x))^6 ).
G.f. A(x) satisfies A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2), where G(x) = 1 + x*G(x)^13.
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_7>=0 and x_1+2*(x_2+x_3+...+x_7)=n-1} a(x_1) * Product_{k=2..7} a(2*x_k). (End)
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_13>=0 and x_1+x_2+...+x_13=n-1} (-1)^(x_1+x_2+x_3+x_4+x_5+x_6) * Product_{k=1..13} a(x_k). - Seiichi Manyama, Jul 09 2025

a(11) onwards from Andrew Howroyd, Jan 26 2024

a(0)=1 prepended by Seiichi Manyama, Jul 07 2025

A385687 E.g.f. A(x) satisfies A(x) = exp( x*((A(x) + A(-x))/2)^2 ).

Original entry on oeis.org

1, 1, 1, 7, 25, 341, 2161, 44115, 404209, 11010025, 132273601, 4508793983, 67085545033, 2747071330173, 48765277295281, 2331905267846731, 48106649137922017, 2631174441142423505, 61862217319644572161, 3809106344377237185399, 100542158725584301036921

Offset: 0

Seiichi Manyama, Jul 06 2025

nonn

Cf. A058014, A385688.

Cf. A143546, A360987, A385690.

terms = 21;  A[] = 1; Do[A[x] = Exp[x*((A[x] + A[-x])/2)^2] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jul 07 2025 *)

E.g.f. A(x) satisfies A(-x) = 1/A(x).

a(0) = 1; a(n) = (n-1)! * Sum_{i, j, k>=0 and i+2*j+2*k=n-1} (n-i) * a(i) * a(2*j) * a(2*k)/(i! * (2*j)! * (2*k)!).

cp's OEIS Frontend

A233669 a(n) = 7*binomial(5*n+7, n)/(5*n+7).

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A233736 a(n) = 8*binomial(5*n + 8, n)/(5*n + 8).

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A233737 a(n) = 9*binomial(5*n+9, n)/(5*n+9).

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A143338 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3*A(-x).

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A192893 Number of symmetric 11-ary factorizations of the n-cycle (1,2...n).

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A369473 Number of chiral pairs of polyominoes composed of n hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}.

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Mathematica

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A192894 Number of symmetric 13-ary factorizations of the n-cycle (1,2...n).

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A385687 E.g.f. A(x) satisfies A(x) = exp( x*((A(x) + A(-x))/2)^2 ).

A233669 a(n) = 7binomial(5n+7, n)/(5*n+7).

A233736 a(n) = 8binomial(5n + 8, n)/(5*n + 8).

A233737 a(n) = 9binomial(5n+9, n)/(5*n+9).

A143338 G.f. A(x) satisfies A(x) = 1 + xA(x)^3A(-x).