A002296 -id:A002296 - cp's OEIS frontend

A235535 a(n) = binomial(9n, 3n) / (6*n + 1).

1, 12, 1428, 246675, 50067108, 11124755664, 2619631042665, 642312451217745, 162250238001816900, 41932353590942745504, 11034966795189838872624, 2946924270225408943665279, 796607831560617902288322405, 217550867863011281855594752680

Offset: 0

Bruno Berselli, Jan 12 2014

This is the case l=6, k=3 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.

Also, the sequence follows A002296 and A235536, namely binomial(7*n,n)/(6*n+1) and binomial(8*n,2*n)/(6*n+1); naturally, even binomial(10*n,4*n)/(6*n+1) is always integer.

Robert Israel, Table of n, a(n) for n = 0..403
Zhi-Wei Sun, On Divisibility Of Binomial Coefficients, Journal of the Australian Mathematical Society 93 (2012), p. 189-201.

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), A235534 (l=4, k=2), A235536 (l=6, k=2), A187357 (l=3, k=3), this sequence (l=6, k=3).

l:=6; k:=3; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* here l is divisible by all the prime factors of k */

seq(binomial(9*n,3*n)/(6*n+1), n=0..30); # Robert Israel, Feb 15 2021

Table[Binomial[9 n, 3 n]/(6 n + 1), {n, 0, 20}]

a(n) = A001764(3*n) = A047749(6*n).
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 6F5(1/9,2/9,4/9,5/9,7/9,8/9; 1/3,1/2,2/3,5/6,7/6; 19683*x/64).
a(n) ~ 3^(9*n-1)/(sqrt(Pi)*4^(3*n+1)*n^(3/2)). (End)
D-finite with recurrence 8*(6*n + 5)*(2*n + 1)*(n + 1)*(3*n + 2)*(3*n + 1)*(6*n + 7)*a(n + 1) = 3*(9*n + 8)*(9*n + 7)*(9*n + 5)*(9*n + 4)*(9*n + 2)*(9*n + 1)*a(n). - Robert Israel, Feb 15 2021

A346938 a(n) = Sum_{d|n} mu(n/d) * binomial(7d,d) / (6d+1).

Original entry on oeis.org

1, 6, 69, 812, 10471, 141702, 1997687, 28988856, 430321563, 6503342378, 99726673129, 1547847703500, 24269405074739, 383846166714410, 6116574500850339, 98106248277869040, 1582638261961640246, 25661404527359789034, 417980115131315136399, 6836064539918615002932

Offset: 1

Ilya Gutkovskiy, Aug 08 2021

nonn

Moebius transform of A002296.

Cf. A002296, A002996, A008683, A346925, A346935, A346936, A346937, A346939.

Table[Sum[MoebiusMu[n/d] Binomial[7 d, d]/(6 d + 1), {d, Divisors[n]}], {n, 20}]

A349590 G.f. A(x) satisfies: A(x) = 1 / (1 - 2x) + x (1 - 2x)^5 A(x)^7.

Original entry on oeis.org

1, 3, 15, 132, 1595, 22134, 329718, 5136028, 82579819, 1359902823, 22818697128, 388728802702, 6705324823466, 116878939752376, 2055505806198352, 36427660285955808, 649894104351874395, 11662729497015257677, 210383830525447606431, 3812719304673511150854

Offset: 0

Ilya Gutkovskiy, Nov 22 2021

nonn

Second binomial transform of A002296.

Cf. A002296, A064613, A346649, A346762, A349581, A349582, A349584, A349591.

nmax = 19; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x)^5 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[7 k, k] 2^(n - k)/(6 k + 1), {k, 0, n}], {n, 0, 19}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(7*k,k)*2^(n-k)/(6*k+1)); \\ Michel Marcus, Nov 23 2021

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(7*k,k) * 2^(n-k) / (6*k+1).
a(n) = 2^n*F([1/7, 2/7, 3/7, 4/7, 5/7, 6/7, -n], [1/3, 1/2, 2/3, 5/6, 1, 7/6], -7^7/(2^7*3^6)), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021
a(n) ~ 916855^(n + 3/2) / (282475249 * sqrt(Pi) * n^(3/2) * 3^(6*n + 3/2) * 4^(3*n + 1)). - Vaclav Kotesovec, Nov 26 2021

A386380 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} a(6k) a(n-1-6*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 15, 24, 34, 45, 57, 70, 154, 253, 368, 500, 650, 819, 1827, 3045, 4495, 6200, 8184, 10472, 23562, 39627, 59052, 82251, 109668, 141778, 320866, 543004, 814506, 1142295, 1533939, 1997688, 4540200, 7718340, 11633440, 16398200, 22137570

Offset: 0

Seiichi Manyama, Jul 20 2025

nonn

Wikipedia, Fuss-Catalan number

Cf. A002296, A233832, A233833, A386392, A233834, A130565.

Cf. A047749, A118968, A124753, A386379, A386396.

A386380 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        add(procname(6*k)*procname(n-1-6*k),k=0..floor((n-1)/6)) ;
    end if;
end proc:
seq(A386380(n),n=0..80) ; # R. J. Mathar, Jul 30 2025

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\6, 7, n%6+1);

For k=0..5, a(6*n+k) = (k+1) * binomial(7*n+k+1,n)/(7*n+k+1).

G.f. A(x) satisfies A(x) = 1/(1 - x * Product_{k=0..5} A(w^k*x)), where w = exp(Pi*i/3).

A235536 a(n) = binomial(8n, 2n) / (6*n + 1).

Original entry on oeis.org

1, 4, 140, 7084, 420732, 27343888, 1882933364, 134993766600, 9969937491420, 753310723010608, 57956002331347120, 4524678117939182220, 357557785658996609700, 28545588568201512137904, 2298872717007844035521848, 186533392975795702301759056

Offset: 0

Bruno Berselli, Jan 12 2014

This is the case l=6, k=2 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.
First bisection of A002293.
Also, the sequence is between A002296 and A235535.

Zhi-Wei Sun, On Divisibility Of Binomial Coefficients, Journal of the Australian Mathematical Society 93 (2012), p. 189-201.

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), A235534 (l=4, k=2), this sequence (l=6, k=2), A187357 (l=3, k=3), A235535 (l=6, k=3).

l:=6; k:=2; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* where l is divisible by all the prime factors of k */

Table[Binomial[8 n, 2 n]/(6 n + 1), {n, 0, 20}]

a(n) = A124753(6*n).
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 6F5(1/8,1/4,3/8,5/8,3/4,7/8; 1/3,1/2,2/3,5/6,7/6; 65536*x/729).
a(n) ~ 2^(16*n-1)/(sqrt(Pi)*3^(6*n+3/2)*n^(3/2)). (End)

A364476 G.f. satisfies A(x) = 1 + xA(x) + x^2A(x)^7.

Original entry on oeis.org

1, 1, 2, 9, 44, 226, 1241, 7093, 41666, 250260, 1529993, 9488398, 59545909, 377451385, 2413157855, 15542535697, 100753850132, 656856027658, 4303970039402, 28328599504756, 187214549485759, 1241775795647609, 8263989319451514, 55163575187733922

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000045, A000108, A001006, A182454, A186996, A364472.

Cf. A002296, A364477.

a(n) = sum(k=0, n\2, binomial(n+5*k, k)*binomial(n+4*k, n-2*k)/(6*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(n+5*k,k) * binomial(n+4*k,n-2*k) / (6*k+1) = Sum_{k=0..floor(n/2)} binomial(n+5*k,7*k) * binomial(7*k,k) / (6*k+1).

A206289 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x*(1 - x^k) ).

Original entry on oeis.org

1, 1, 2, 4, 10, 25, 73, 214, 679, 2189, 7331, 24867, 86269, 302144, 1072621, 3837768, 13853674, 50319789, 183941789, 675731105, 2494370326, 9244865453, 34394851701, 128390336942, 480749791772, 1805161153783, 6795744287172, 25643914891284, 96980809856731

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Compare to the g.f. of partitions of n into distinct parts (A000009): Sum_{n>=0} Product_{k=1..n} x*(1 + x^k).

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 73*x^6 + 214*x^7 +...
such that, by definition,
A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
where G_n( x*(1 - x^n) ) = x.
The first few expansions of G_n(x) begin:
G_1(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 +...+ A000108(n)*x^(n+1) +...
G_2(x) = x + x^3 + 3*x^5 + 12*x^7 + 55*x^9 +...+ A001764(n)*x^(2*n+1) +...
G_3(x) = x + x^4 + 4*x^7 + 22*x^10 + 140*x^13 +...+ A002293(n)*x^(3*n+1) +...
G_4(x) = x + x^5 + 5*x^9 + 35*x^13 + 285*x^17 +...+ A002294(n)*x^(4*n+1) +...
G_5(x) = x + x^6 + 6*x^11 + 51*x^16 + 506*x^21 +...+ A002295(n)*x^(5*n+1) +...
G_6(x) = x + x^7 + 7*x^13 + 70*x^19 + 819*x^25 +...+ A002296(n)*x^(6*n+1) +...
Note that G_n(x) = x + x*G_n(x)^(n+1).

Vaclav Kotesovec, Table of n, a(n) for n = 0..260

Cf. A206290, A194560.

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,serreverse(x*(1-x^k+x*O(x^n))))),n)}
for(n=0,35,print1(a(n),", "))

G.f.: Sum_{n>=0} Product_{k=1..n} G_k(x), where G_n(x) is defined by:
(1) G_n(x) = Series_Reversion( x*(1 - x^n) ),
(2) G_n(x) = x + x*G_n(x)^(n+1),
(3) G_n(x) = Sum_{k>=0} binomial(n*k+k+1, k) * x^(n*k+1) / (n*k+k+1).
a(n) ~ c * 4^n / n^(3/2), where c = 0.19197348199... . - Vaclav Kotesovec, Nov 06 2014

A059967 Number of 9-ary trees.

Original entry on oeis.org

1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, 3573805950, 70625252863, 1416298046436, 28748759731965, 589546754316126, 12195537924351375, 254184908607118800, 5332692942907262361, 112524941404978156215

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Mar 05 2001

nonn

S. Heubach, N. Y. Li and T. Mansour, Staircase tilings and k-Catalan structures, Discrete Math., 308 (2008), 5954-5964.
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

Related algebraic sequences concerning trees: strictly k-ary trees (A000108: s=x+s^2, A001263: s=(x, y)+(x, s)+(s, y)+(s, s))), (A001764: s=x+s^3), (A002293: s=x+s^4), (A002294: s=x+s^5), (A002295: s=x+s^6), (A002296: s=x+s^7), (A007556: s=x+s^8), at most k-ary trees (A001006: s=x+xs+xs^2), (A036765-A036769, s=x+xs^2....+xs^k, k=3, 4, 5, 6, 7).

with(combinat): for n from 1 to 40 do printf(`%d,`,binomial(9*n,n)/((9-1)*n+1)) od:

G.f. A(x) satisfies: A = x + A^9.

a(n) = C(k*n, n)/((k-1)*n+1), k=9.

More terms from James Sellers, Mar 15 2001

A386392 a(n) = 4 * binomial(7n+4,n)/(7n+4).

Original entry on oeis.org

1, 4, 34, 368, 4495, 59052, 814506, 11633440, 170574723, 2552698720, 38832808586, 598724403680, 9335085772194, 146936230074004, 2331703871687400, 37263447339612480, 599206511767593099, 9688121925389895636, 157401957319775436400, 2568427016865897264000

Offset: 0

Seiichi Manyama, Jul 20 2025

nonn

Wikipedia, Fuss-Catalan number

Cf. A118971, A212073, A234463, A234507, A234527, A234870.

Cf. A002296, A386380.

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n, 7, 4);

a(n) = r * binomial(n*p+r,n)/(n*p+r), the Fuss-Catalan number with p=7 and r=4.
a(n) = A386380(6*n+3).
G.f. A(x) satisfies A(x) = (1 + x*A(x)^(p/r))^r, where p=7, r=4.
G.f.: B(x)^4, where B(x) is the g.f. of A002296.

A137211 Generalized or s-Catalan numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 5, 12, 22, 1, 14, 55, 140, 285, 1, 42, 273, 969, 2530, 5481, 1, 132, 1428, 7084, 23751, 62832, 141778, 1, 429, 7752, 53820, 231880, 749398, 1997688, 4638348, 1, 1430, 43263, 420732, 2330445, 9203634, 28989675, 77652024

Offset: 1

Roger L. Bagula, Mar 05 2008

nonn

From R. J. Mathar, May 04 2008: (Start)
This is a triangular section of Stanica's array of s-Catalan numbers, with rows A000108, A001764, A002293-A002296, A007556, A062994, A059968,... read along diagonals in A062993 and A070914:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ...
1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, ...
1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, ...
1, 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, ...
1, 1, 6, 51, 506, 5481, 62832, 749398, 9203634, 115607310, 1478314266, ...
1, 1, 7, 70, 819, 10472, 141778, 1997688, 28989675, 430321633, 6503352856, ...
1, 1, 8, 92, 1240, 18278, 285384, 4638348, 77652024, 1329890705, 23190029720, ...
1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, 3573805950, 70625252863, ...
1, 1, 10, 145, 2470, 46060, 910252, 18730855, 397089550, 8612835715, 190223180840, ...
(End)
The Fuss-Catalan numbers are Cat(d,k)= [1/(k*(d-1)+1)]*binomial(k*d,k) and enumerate the number of (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Whieldon and Schuetz link for this interpretation and others), so the (k+1)-th column of Stanica's array enumerates the number of (n+1)-gon partitions of a (k*(n-1)+2)-gon. Cf. A000326 (k=3), A100157 (k=4) and A234043 (k=5). - Tom Copeland, Oct 05 2014

			{1},
{1, 1},
{1, 2, 3},
{1, 5, 12, 22},
{1, 14, 55, 140, 285},
{1, 42, 273, 969, 2530, 5481},
{1, 132, 1428, 7084, 23751, 62832, 141778},
{1, 429, 7752, 53820, 231880, 749398, 1997688, 4638348}

Heinrich Niederhausen, Catalan Traffic at the Beach, Electronic Journal of Combinatorics, Volume 9 (2002), #R33.
A. Regev, The Central Component of a Triangulation, J. Int. Seq. 16 (2013) #13.4.1
Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
P. Stanica, p^q-Catalan numbers and squarefree binomial coefficients, J. Numb. Theory 100 (2003) 203-216.

t[n_, m_] := Binomial[m*n, n]/((m - 1)*n + 1); a = Table[Table[t[n, m], {m, 1, n + 1}], {n, 0, 10}]; Flatten[a]

T(n,m) = binomial(m*n,n)/((m-1)*n+1).

Edited by N. J. A. Sloane, May 16 2008

cp's OEIS Frontend

A235535 a(n) = binomial(9*n, 3*n) / (6*n + 1).

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A346938 a(n) = Sum_{d|n} mu(n/d) * binomial(7*d,d) / (6*d+1).

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A349590 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^5 * A(x)^7.

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A386380 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} a(6*k) * a(n-1-6*k).

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A235536 a(n) = binomial(8*n, 2*n) / (6*n + 1).

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A364476 G.f. satisfies A(x) = 1 + x*A(x) + x^2*A(x)^7.

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A206289 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x*(1 - x^k) ).

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A059967 Number of 9-ary trees.

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A235535 a(n) = binomial(9n, 3n) / (6*n + 1).

A346938 a(n) = Sum_{d|n} mu(n/d) * binomial(7d,d) / (6d+1).

A349590 G.f. A(x) satisfies: A(x) = 1 / (1 - 2x) + x (1 - 2x)^5 A(x)^7.

A386380 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} a(6k) a(n-1-6*k).

A235536 a(n) = binomial(8n, 2n) / (6*n + 1).

A364476 G.f. satisfies A(x) = 1 + xA(x) + x^2A(x)^7.

A386392 a(n) = 4 * binomial(7n+4,n)/(7n+4).