A004343 -id:A004343 - cp's OEIS frontend

A004319 a(n) = binomial(3*n, n - 1).

1, 6, 36, 220, 1365, 8568, 54264, 346104, 2220075, 14307150, 92561040, 600805296, 3910797436, 25518731280, 166871334960, 1093260079344, 7174519270695, 47153358767970, 310325523515700, 2044802197953900, 13488561475572645, 89067326568860640, 588671286046028640

Offset: 1

N. J. A. Sloane

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Milan Janjic, Two Enumerative Functions.
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Cf. A045721 (k=1), A025174 (k=2), A236194 (k=4), A013698 (k=5), A165817 (k=-1), A117671 (k=-2).

Cf. A000139, A001764, A005809, A006013, A236194, A001791, A004331, A004343, A004356, A004369, A004382.

A004319 := proc(n)
binomial(3*n,n-1);
end proc: # R. J. Mathar, Aug 10 2015

Table[Binomial[3n, n - 1], {n, 20}] (* Harvey P. Dale, Sep 21 2011 *)

a(n):=sum((binomial(3*i-1,2*i-1)*binomial(3*n-3*i-3,2*n-2*i-2))/(2*n-2*i-1),i,1,n-1)/2; /* Vladimir Kruchinin, May 15 2013 */

vector(30, n, binomial(3*n, n-1)) \\ Altug Alkan, Nov 04 2015

G.f.: (g-1)/(1-3*z*g^2), where g = g(z) is given by g = 1 + z*g^3, g(0) = 1, i.e. (in Maple notation), g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003
a(n) = Sum_{i=0..n-1} binomial(i+2*n, i). - Ralf Stephan, Jun 03 2005
D-finite with recurrence -2*(2*n+1)*(n-1)*a(n) + 3*(3*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013
a(n) = (1/2) * Sum_{i=1..n-1} binomial(3*i - 1, 2*i - 1)*binomial(3*n - 3*i - 3, 2*n - 2*i - 2)/(2*n - 2*i - 1). - Vladimir Kruchinin, May 15 2013
G.f.: x*hypergeom2F1(5/3, 4/3; 5/2; 27x/4). - R. J. Mathar, Aug 10 2015
a(n) = n*A001764(n). - R. J. Mathar, Aug 10 2015
From Peter Bala, Nov 04 2015: (Start)
With offset 0, the o.g.f. equals f(x)*g(x)^3, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k, n). See the cross-references. (End)
G.f.: cos(t)/(2*sqrt(1 - (27*x)/4)) - sin(t)/(sqrt(3)*sqrt(x)), where t = arcsin((sqrt(27*x))/2)/3. - Vladimir Kruchinin, May 13 2016
a(n) = [x^(2*n+1)] 1/(1 - x)^n. - Ilya Gutkovskiy, Oct 10 2017
a(n) = binomial(n+1, 2) * A000139(n). - F. Chapoton, Feb 23 2024

A004331 Binomial coefficient C(4n,n-1).

Original entry on oeis.org

1, 8, 66, 560, 4845, 42504, 376740, 3365856, 30260340, 273438880, 2481256778, 22595200368, 206379406870, 1889912732400, 17345898649800, 159518999862720, 1469568786235308, 13559593014190944, 125288932441604200

Offset: 1

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A002293, A005810, A052203, A224274, A257633, A262977.

Cf. A001791, A004319, A004343, A004356, A004369, A004382.

#A004331
seq(binomial(4*n - 1,n), n = 0..20);

a[n_] := Binomial[4*n, n - 1]; Array[a, 19] (* Amiram Eldar, May 09 2020 *)

vector(30, n, binomial(4*n, n-1)) \\ Altug Alkan, Nov 05 2015

G.f.: (g^2-g)/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011
With an offset of 0, the o.g.f. equals f(x)*g(x)^4, where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A262977 (k = -1), A005810 (k = 0), A052203 (k = 1), A257633 (k = 2) and A224274 (k = 3). - Peter Bala, Nov 04 2015
D-finite with recurrence 3*(n-1)*(3*n-1)*(3*n+1)*a(n) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Mar 19 2025

A365341 a(n) = (5n)!/(4n+1)!.

Original entry on oeis.org

1, 1, 10, 210, 6840, 303600, 17100720, 1168675200, 93963542400, 8691104822400, 909171781056000, 106137499051584000, 13679492361575040000, 1929327666754295808000, 295570742023171270656000, 48877281133334949335040000, 8677556868736487617966080000

Offset: 0

Seiichi Manyama, Sep 01 2023

Cf. A001761, A001763, A052795, A365340.
Cf. A004343.
Cf. A000142, A002294.

PARI
```
a(n) = (5*n)!/(4*n+1)!;
```

from sympy import ff
def A365341(n): return ff(5*n,n-1) # Chai Wah Wu, Sep 01 2023

E.g.f.: exp( 1/5 * Sum_{k>=1} binomial(5*k,k) * x^k/k ). - Seiichi Manyama, Feb 08 2024
a(n) = A000142(n)*A002294(n). - Alois P. Heinz, Feb 08 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^4).
a(n) = Sum_{k=0..n} (4*n+1)^(k-1) * |Stirling1(n,k)|. (End)

A023838 Sum of exponents in prime-power factorization of C(5n,n-1).

Original entry on oeis.org

0, 2, 3, 5, 5, 7, 8, 9, 8, 10, 10, 12, 14, 13, 15, 14, 14, 16, 14, 17, 17, 18, 18, 21, 18, 22, 24, 23, 23, 24, 24, 24, 23, 23, 24, 25, 24, 26, 28, 29, 25, 30, 27, 29, 33, 30, 30, 32, 32, 33, 33, 37, 37, 38, 38, 37, 36, 40, 39, 44, 39, 41, 40, 39, 37, 39, 38, 39, 42, 41, 44, 44, 43, 43, 44, 46, 48, 49

Offset: 1

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A001222, A004343, A023837, A023839, A023840, A023841, A023842, A023843, A023844, A023845, A023846, A023847, A023848, A023849, A023850, A023851.

Join[{0},Table[Total[Transpose[FactorInteger[Binomial[5 n,n-1]]] [[2]]],{n,2,80}]] (* Harvey P. Dale, Dec 31 2012 *)
a[n_] := PrimeOmega[Binomial[5*n, n-1]]; Array[a, 100] (* Amiram Eldar, Jun 14 2025 *)

a(n) = bigomega(binomial(5*n,n-1)); \\ Amiram Eldar, Jun 14 2025

From Amiram Eldar, Jun 14 2025: (Start)
a(n) = A001222(A004343(n)).
a(n) = A023837(n) - A001222(4*n+1) + A001222(n). (End)

A004356 Binomial coefficient C(6n,n-1).

Original entry on oeis.org

1, 12, 153, 2024, 27405, 376992, 5245786, 73629072, 1040465790, 14783142660, 210980549208, 3022285436352, 43430966148115, 625806790696080, 9038619861406740, 130815226545884704, 1896724514424115530

Offset: 1

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Seiichi Manyama, Table of n, a(n) for n = 1..853
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A001791, A004319, A004331, A004343, A004369, A004382.

Table[Binomial[6n,n-1],{n,20}] (* Harvey P. Dale, Mar 04 2017 *)

D-finite with recurrence 5*(n-1)*(5*n+1)*(5*n-3)*(5*n-2)*(5*n-1)*a(n) -72*(6*n-5)*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Mar 19 2025

A004369 Binomial coefficient C(7n,n-1).

Original entry on oeis.org

1, 14, 210, 3276, 52360, 850668, 13983816, 231917400, 3872894697, 65033528560, 1096993404430, 18574174153080, 315502265971620, 5373846361969456, 91748617512913200, 1569699972909739440

Offset: 1

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Seiichi Manyama, Table of n, a(n) for n = 1..804
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A001791, A004319, A004331, A004343, A004356, A004382.

cp's OEIS Frontend

A004319 a(n) = binomial(3*n, n - 1).

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A004331 Binomial coefficient C(4n,n-1).

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A365341 a(n) = (5*n)!/(4*n+1)!.

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A023838 Sum of exponents in prime-power factorization of C(5n,n-1).

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Mathematica

PARI

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A004356 Binomial coefficient C(6n,n-1).

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Mathematica

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A004369 Binomial coefficient C(7n,n-1).

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A365341 a(n) = (5n)!/(4n+1)!.