A130564 -id:A130564 - cp's OEIS frontend

A212072 G.f. satisfies: A(x) = (1 + x*A(x)^2)^3.

1, 3, 21, 190, 1950, 21576, 250971, 3025308, 37456650, 473498025, 6085977381, 79296104784, 1044955576496, 13903071489300, 186507160795350, 2519857658331576, 34258270557555282, 468322722628414290, 6433538749783033350, 88767899653496377050, 1229626632793564911906

Offset: 0

Paul D. Hanna, Apr 29 2012

nonn

			G.f.: A(x) = 1 + 3*x + 21*x^2 + 190*x^3 + 1950*x^4 + 21576*x^5 + ...
Related expansions:
A(x)^2 = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 + ... + A002295(n+1)*x^n + ...
A(x)^(1/3) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + ... + A002295(n)*x^n + ...

Michael De Vlieger, Table of n, a(n) for n = 0..855
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
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.

Cf. A002295, A212071, A212073, A130564.

Table[(3 Binomial[#, n])/# &[6 n + 3], {n, 0, 20}] (* Michael De Vlieger, May 13 2022 *)

{a(n)=binomial(6*n+3,n) * 3/(6*n+3)}
for(n=0, 40, print1(a(n), ", "))

{a(n)=local(A=1+3*x); for(i=1, n, A=(1+x*A^2)^3+x*O(x^n)); polcoeff(A, n)}

a(n) = 3*binomial(6*n+3,n)/(6*n+3).

G.f. A(x) = G(x)^3 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

A235340 a(n) = 10binomial(11n+10,n)/(11*n+10).

Original entry on oeis.org

1, 10, 155, 2870, 58565, 1270752, 28765650, 671650110, 16057800980, 391139588190, 9672348219898, 242182964452000, 6127720969229265, 156431295179478200, 4024231652469275640, 104218796026870015374, 2714941275486017847825

Offset: 0

Tim Fulford, Jan 06 2014

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=10.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007; Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A130564, A230388, A234868, A234869, A234870, A234871, A234872, A234873, A235338, A235339.

[10*Binomial(11*n+10, n)/(11*n+10): n in [0..30]];

Table[10 Binomial[11 n + 10, n]/(11 n + 10), {n, 0, 30}]

a(n) = 10*binomial(11*n+10,n)/(11*n+10);

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/10))^10+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=11, r=10.
From Wolfdieter Lang, Feb 15 2024: (Start)
a(n) = binomial(11*n + 9, n + 1)/(10*n + 9) which is instance k = 10 of c(k, n+1) given in a comment in A130564.
x*B(x), with the g.f. above named B(x), is the compositional inverse of y*(1 - y)^10, hence  B(x)*(1 - x*B(x))^10 = 1.
G.f.: 11F10([10..20]/11, [11..20]/10; (11^11/10^10)*x) =  (10/(11*x))*(1 - 10F9([-1,1,2,3,4,5,6,7,8,9]/11, [1,2,3,4,5,6,7,8,9]/10; (11^11/10^10)*x)).
 (End)

A386368 a(n) = Sum_{k=0..n-1} binomial(6k,k) binomial(6n-6k-2,n-k-1).

Original entry on oeis.org

0, 1, 16, 246, 3736, 56421, 849432, 12763878, 191548464, 2871970110, 43031833656, 644432826478, 9646983339456, 144366433138955, 2159869510669320, 32306874783230556, 483151884326658144, 7224464127509984490, 108011596038055519680, 1614676987907480393940

Offset: 0

Seiichi Manyama, Jul 19 2025

nonn

			(1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ) = x + 8*x^2 + 82*x^3 + 934*x^4 + 56421*x^5/5 + ...

Cf. A000302, A036829, A308523, A386367.
Cf. A079679, A386567, A386615, A386616.
Cf. A002295, A130564.

A386368 := proc(n::integer)
    add(binomial(6*k,k)*binomial(6*n-6*k-2,n-k-1),k=0..n-1) ;
end proc:
seq(A386368(n),n=0..80) ; # R. J. Mathar, Jul 30 2025

a(n) = sum(k=0, n-1, binomial(6*k, k)*binomial(6*n-6*k-2, n-k-1));

my(N=20, x='x+O('x^N), g=x*sum(k=0, N, binomial(6*k+4, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-6*g)^2))

G.f.: g*(1-g)/(1-6*g)^2 where g*(1-g)^5 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ).
G.f.: (g-1)/(6-5*g)^2 where g=1+x*g^6.
a(n) = Sum_{k=0..n-1} binomial(6*k-2+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n-1,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k-1,k).
Conjecture D-finite with recurrence 48828125*(n-1)*(5*n-4)*(5*n-3) *(432862082629612805*n -769306661967834399) *(5*n-2)*(5*n-1)*a(n) +1125000*(-405245406115816219575000*n^6 +2613180799468910510392500*n^5 -7667164406968651479521250*n^4 +13834502135358262506660375*n^3 -16251583347734702117341345*n^2 +11251247074043948959380314*n -3395699069351241765495720)*a(n-1) +33592320*(142281690918326440537500*n^6 -1266424338521609272012500*n^5 +5236041263583271687953750*n^4 -12786608152035075786775875*n^3 +18838556229131595646260055*n^2 -15323925851720394901667853*n +5240681406952416812161236)*a(n-2) -53444359913472*(6*n-17) *(395547729523405*n -538181211711288)*(6*n-13) *(3*n-7)*(2*n-5) *(3*n-8)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A371519 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^5.

Original entry on oeis.org

1, 5, 45, 470, 5375, 65231, 825225, 10764185, 143739440, 1955340360, 27001732972, 377530388235, 5333865386885, 76031188364860, 1092117166466660, 15792298241897649, 229704197116753825, 3358528175751886765, 49333470827844265285, 727680248026484478405

Offset: 0

Seiichi Manyama, Mar 26 2024

nonn

Cf. A349333, A371379, A371521, A371522, A371523.
Cf. A270386, A371483, A371486.
Cf. A130564.

a(n) = sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+4, k)/(k+1));

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+4,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^5 ).
G.f.: A(x) = B(x)^5 where B(x) is the g.f. of A349333.

A386567 a(n) = Sum_{k=0..n-1} binomial(6k-1,k) binomial(6n-6k,n-k-1).

Original entry on oeis.org

0, 1, 17, 268, 4129, 62955, 954392, 14417376, 217279857, 3269099590, 49125066135, 737516631908, 11064270530632, 165889863957065, 2486052264852180, 37241727274394640, 557707191712371729, 8349517132932620730, 124971965902300790390, 1870139909398530770760

Offset: 0

Seiichi Manyama, Jul 26 2025

nonn

			(1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ) = x + 17*x^2/2 + 268*x^3/3 + 4129*x^4/4 + 12591*x^5 + ...

Cf. A000346, A062236, A386565, A386566.
Cf. A079679, A386368, A386615, A386616.
Cf. A002295, A130564.

a(n) = sum(k=0, n-1, binomial(6*k-1, k)*binomial(6*n-6*k, n-k-1));

my(N=20, x='x+O('x^N), g=sum(k=0, N, binomial(6*k, k)/(5*k+1)*x^k)); concat(0, Vec(g*(g-1)/(6-5*g)^2))

G.f.: g*(g-1)/(6-5*g)^2 where g=1+x*g^6.
G.f.: g/(1-6*g)^2 where g*(1-g)^5 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(6*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(6*k-1+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k,k).

A233827 a(n) = 8binomial(6n+8,n)/(6*n+8).

Original entry on oeis.org

1, 8, 76, 800, 8990, 105672, 1283464, 15981504, 202927725, 2617624680, 34206162848, 451872681728, 6024664312030, 80964348872400, 1095590286231120, 14915165412813184, 204140673966231870, 2807362363541687280, 38772186055550141700

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=6, 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, A002295, A212071, A212072, A212073, A130564, A233743, A233829, A233830.

[8*Binomial(6*n+8, n)/(6*n+8): n in [0..30]];

Table[8 Binomial[6 n + 8, n]/(6 n + 8), {n, 0, 30}]

PARI
```
a(n) = 8*binomial(6*n+8,n)/(6*n+8);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(6/8))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=6, r=8.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 5F5(4/3,3/2,5/3,11/6,13/6; 1,9/5,11/5,12/5,13/5; 46656*x/3125).
a(n) ~ 3^(6*n+15/2)*4^(3*n+5)/(sqrt(Pi)*5^(5*n+17/2)*n^(3/2)). (End)
D-finite with recurrence 5*n*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+4)*a(n) -72*(6*n+5)*(3*n+2)*(2*n+1)*(3*n+1)*(6*n+7)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A233829 a(n) = 3binomial(6n+9,n)/(2*n+3).

Original entry on oeis.org

1, 9, 90, 975, 11160, 132867, 1629012, 20430900, 260907075, 3381098545, 44352058608, 587787511779, 7858257798300, 105855415586550, 1435361957277480, 19576154604317304, 268364706225271110, 3695862686045572350, 51108790709588823150

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=6, 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, A002295, A212071, A212072, A212073, A130564, A233743, A233827, A233830.

[3*Binomial(6*n+9, n)/(2*n+3): n in [0..30]];

Table[3 Binomial[6 n + 9, n]/(2 n + 3), {n, 0, 30}]

PARI
```
a(n) = 3*binomial(6*n+9,n)/(2*n+3);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(2/3))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=6, r=9.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 5F5(3/2,5/3,11/6,13/6,7/3; 1,11/5,12/5,13/5,14/5; 46656*x/3125).
a(n) ~ 3^(6*n+21/2)*4^(3*n+4)/(sqrt(Pi)*5^(5*n+19/2)*n^(3/2)). (End)

A233830 a(n) = 5binomial(6n+10,n)/(3*n+5).

Original entry on oeis.org

1, 10, 105, 1170, 13640, 164502, 2036265, 25727800, 330482295, 4303216330, 56672074888, 753573733050, 10103474312100, 136435868978220, 1854009194816745, 25333847134998864, 347880174736462550, 4798137522234602700, 66441427922465470095, 923346006310186106010, 12873823246049001482400

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=6, 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.

Cf. A000108, A002295, A212071, A212072, A212073, A130564, A233743, A233827, A233829.

[5*Binomial(6*n+10, n)/(3*n+5): n in [0..30]];

Table[5 Binomial[6 n + 10, n]/(3 n + 5), {n, 0, 30}]

PARI
```
a(n) = 5*binomial(6*n+10,n)/(3*n+5);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/5))^10+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=6, r=10.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 6F6(5/3,11/6,2,13/6,7/3,5/2; 1,11/5,12/5,13/5,14/5,3; 46656*x/3125).
a(n) ~ 3^(6*n+19/2)*4^(3*n+5)/(sqrt(Pi)*5^(5*n+19/2)*n^(3/2)). (End)

A386379 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/5)} a(5k) a(n-1-5*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 13, 21, 30, 40, 51, 114, 190, 280, 385, 506, 1150, 1950, 2925, 4095, 5481, 12586, 21576, 32736, 46376, 62832, 145299, 250971, 383838, 548340, 749398, 1741844, 3025308, 4654320, 6690585, 9203634, 21475146, 37456650, 57887550

Offset: 0

Seiichi Manyama, Jul 20 2025

nonn

Wikipedia, Fuss-Catalan number

Cf. A002295, A212071, A212072, A212073, A130564.

Cf. A047749, A118968, A124753, A386380.

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\5, 6, n%5+1);

For k=0..4, a(5*n+k) = (k+1) * binomial(6*n+k+1,n)/(6*n+k+1).

G.f. A(x) satisfies A(x) = 1/(1 - x * Product_{k=0..4} A(w^k*x)), where w = exp(2*Pi*i/5).

A386558 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = binomial((k+1)*n+k-1,n)/(n+1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 5, 0, 1, 4, 15, 30, 14, 0, 1, 5, 26, 91, 143, 42, 0, 1, 6, 40, 204, 612, 728, 132, 0, 1, 7, 57, 385, 1771, 4389, 3876, 429, 0, 1, 8, 77, 650, 4095, 16380, 32890, 21318, 1430, 0, 1, 9, 100, 1015, 8184, 46376, 158224, 254475, 120175, 4862, 0

Offset: 0

Seiichi Manyama, Jul 26 2025

			Square array begins:
  1,   1,    1,     1,      1,      1,       1, ...
  0,   1,    2,     3,      4,      5,       6, ...
  0,   2,    7,    15,     26,     40,      57, ...
  0,   5,   30,    91,    204,    385,     650, ...
  0,  14,  143,   612,   1771,   4095,    8184, ...
  0,  42,  728,  4389,  16380,  46376,  109668, ...
  0, 132, 3876, 32890, 158224, 548340, 1533939, ...

Wikipedia, Fuss-Catalan number

Columns k=0..10 give A000007, A000108, A006013, A006632, A118971, A130564(n+1), A130565(n+1), A234466, A234513, A234573, A235340.
Main diagonal gives A177784(n+1).
Cf. A162382.

a(n, k) = binomial((k+1)*n+k-1, n)/(n+1);

For k > 0, A(n,k) = r * binomial(n*p+r,n)/(n*p+r), the Fuss-Catalan number with p=k+1 and r=k.

G.f. of column k: (1/x) Series_Reverion( x*(1-x)^k ).

cp's OEIS Frontend

A212072 G.f. satisfies: A(x) = (1 + x*A(x)^2)^3.

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A235340 a(n) = 10*binomial(11*n+10,n)/(11*n+10).

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A386368 a(n) = Sum_{k=0..n-1} binomial(6*k,k) * binomial(6*n-6*k-2,n-k-1).

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A371519 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^5.

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A386567 a(n) = Sum_{k=0..n-1} binomial(6*k-1,k) * binomial(6*n-6*k,n-k-1).

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A233827 a(n) = 8*binomial(6*n+8,n)/(6*n+8).

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A233829 a(n) = 3*binomial(6*n+9,n)/(2*n+3).

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A235340 a(n) = 10binomial(11n+10,n)/(11*n+10).

A386368 a(n) = Sum_{k=0..n-1} binomial(6k,k) binomial(6n-6k-2,n-k-1).

A386567 a(n) = Sum_{k=0..n-1} binomial(6k-1,k) binomial(6n-6k,n-k-1).

A233827 a(n) = 8binomial(6n+8,n)/(6*n+8).

A233829 a(n) = 3binomial(6n+9,n)/(2*n+3).

A233830 a(n) = 5binomial(6n+10,n)/(3*n+5).

A386379 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/5)} a(5k) a(n-1-5*k).