A006588 -id:A006588 - cp's OEIS frontend

A133402 Hankel transform of A006588.

1, 96, 405504, 77007421440, 661630022502580224, 257876005135691663309537280, 4565900567740737406606787243126292480, 3675506444195600567841408683430769715388692299776

Offset: 0

Paul Barry, Jan 08 2008

a(n)=4^(n(n+1))*A127636(n); a(n)=A000244(n)*A051255(n)*A053765(n+1);

A228484 a(n) = 2^n(3n)!/(n!(2n)!).

Original entry on oeis.org

1, 6, 60, 672, 7920, 96096, 1188096, 14883840, 188280576, 2399654400, 30766095360, 396363202560, 5126871859200, 66538909237248, 866061993246720, 11300615801536512, 147773778404769792, 1936073567335219200, 25408660721789829120, 333963051307735449600

Offset: 0

Johannes W. Meijer, Aug 22 2013

Oblique diagonal of the Pell-Jacobsthal triangle A013609. Its mirror diagonal is A006588.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A005809, A013609, A006588, A137209.

[2^n*Factorial(3*n)/(Factorial(n)*Factorial(2*n)): n in [0..20]]; // Vincenzo Librandi, Aug 24 2013

a := n -> 2^n*binomial(3*n, n): seq(a(n), n=0..16);

Table[2^n (3 n)!/(n! (2 n)!), {n, 0, 20}] (* Vincenzo Librandi, Aug 24 2013 *)

a(n) = 2^n*binomial(3*n, 2*n); \\ Michel Marcus, Mar 06 2022

a(n) = 2^n*A005809(n).
a(n) = A013609(3*n, n).
a(n) = A006588(n)/2^n.
a(n) = (2*n+1)*A153231(n).
Asymptotic approximation of a(n) ~ C*(13.5)^n/sqrt(n) with C = (1/2)*sqrt(3/Pi) = A137209.
Sum_{n>=0} 1/a(n) = (11*Pi - 12*log(2) + 270)/250. - Amiram Eldar, Mar 06 2022
From Karol A. Penson, Feb 26 2025: (Start)
G.f.:   hypergeom([1/3,2/3],[1/2],27*z/2).
E.g.f.: hypergeom([1/3,2/3],[1/2,1],27*z/2). (End)

More terms from Vincenzo Librandi, Aug 24 2013

A006587 a(n) = 32^(2n)(3n)!/((2n)!n!).

Original entry on oeis.org

3, 36, 720, 16128, 380160, 9225216, 228114432, 5715394560, 144599482368, 3685869158400, 94513444945920, 2435255516528640, 62999001405849600, 1635260233414606848, 42568679092062781440, 1110895735754245275648

Offset: 0

N. J. A. Sloane

W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 35.

Delbert L. Johnson, Table of n, a(n) for n = 0..699
M. Le Brun, Email to N. J. A. Sloane, Jul 1991

Cf. A006588.

A006587:= func< n | 3*4^n*Binomial(3*n,n) >;
[A006587(n): n in [0..40]]; // G. C. Greubel, Aug 27 2025

A006587:=n->3*2^(2*n)*(3*n)!/((2*n)!*n!); seq(A006587(n), n=0..50); # Wesley Ivan Hurt, Nov 23 2013

Table[3*2^(2n)(3n)!/((2n)!*n!), {n, 0, 50}] (* Wesley Ivan Hurt, Nov 23 2013 *)

a(n)=3*binomial(3*n,n)*4^n \\ Charles R Greathouse IV, Aug 11 2017

def A006587(n): return 3*4**n*binomial(3*n,n)
print([A006587(n) for n in range(41)]) # G. C. Greubel, Aug 27 2025

From G. C. Greubel, Aug 27 2025: (Start)
a(n) = 3 * A006588(n).
G.f.: 3*hypergeometric2F1([1/3, 2/3], [1/2], 27*x) = (3/(2*(1-27*x))*( cos(t) + cos(2*t) ), where t = (1/3)*arccos(1-54*x).
E.g.f.: 3*hypergeometric2F2([1/3, 2/3], [1/2, 1], 27*x). (End)

A137209 Decimal expansion of (1/2)*sqrt(3/Pi).

Original entry on oeis.org

4, 8, 8, 6, 0, 2, 5, 1, 1, 9, 0, 2, 9, 1, 9, 9, 2, 1, 5, 8, 6, 3, 8, 4, 6, 2, 2, 8, 3, 8, 3, 4, 7, 0, 0, 4, 5, 7, 5, 8, 8, 5, 6, 0, 8, 1, 9, 4, 2, 2, 7, 7, 0, 2, 1, 3, 8, 2, 4, 3, 1, 5, 7, 4, 4, 5, 8, 4, 1, 0, 0, 0, 3, 6, 1, 6, 3, 6, 5, 3, 0, 4, 0, 5, 6, 1, 4, 8, 1, 8, 7, 0, 3, 9, 7, 0, 0, 4, 2, 4, 1, 5, 7, 6, 4

Offset: 0

Zak Seidov, Mar 05 2008

Decimal expansion of the radius x (in units of cube edge length) of sphere with volume x (in units of cube volume).

Appears in the asymptotic expansions of A228484 and A006588. - Johannes W. Meijer, Aug 22 2013

			0.488602511902919921586384622

Amir Behrouzi-Far, and Doron Zeilberger, On the Average Maximal Number of Balls in a Bin Resulting from Throwing r Balls into n Bins T times, arXiv:1905.07827 [math.CO], 2019.
Marcus Michelen, A Short Note on the Average Maximal Number of Balls in a Bin, Journal of Integer Sequences, Vol. 23 (2020), Article 20.1.7. See C 3,1 Table 2 p. 6. And also on arXiv, arXiv:1905.08933 [math.CO], 2019.
Index entries for transcendental numbers

Cf. A135691.

RealDigits[1/2 Sqrt[3/Pi],10,120][[1]] (* Harvey P. Dale, Jul 11 2017 *)

sqrt(3)/(2*sqrt(Pi)) \\ Michel Marcus, Jun 05 2020

A378779 a(n) = n^2 * 4^n * binomial(3*n, n).

Original entry on oeis.org

0, 12, 960, 48384, 2027520, 76876800, 2737373184, 93351444480, 3084788957184, 99518467276800, 3150448164864000, 98221972499988480, 3023952067480780800, 92119659815689519104, 2781153700681435054080, 83317180181568395673600, 2479232599432958230659072, 73338095004533174933913600

Offset: 0

Amiram Eldar, Dec 07 2024

Amiram Eldar, Table of n, a(n) for n = 0..500
Necdet Batir, On the series Sum_{k=1..oo} binomial(3k,k)^{-1} k^{-n} x^k, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4 (2005), pp. 371-381; arXiv preprint, arXiv:math/0512310 [math.AC], 2005.

Cf. A005809, A006588, A128782.

a[n_] := n^2 * 4^n * Binomial[3*n, n]; Array[a, 25, 0]

PARI
```
a(n) = n^2 * 4^n * binomial(3*n, n);
```

a(n) = A128782(n) * A005809(n).
a(n) = n^2 * A006588(n).
a(n) == 0 (mod 12).
Sum_{n>=1} (-1)^n/a(n) = 6 * arccot(2*sqrt(3)+sqrt(7))^2 - log(2)^2/2 (Batir, 2005, p. 379, eq. (3.8)).

cp's OEIS Frontend

A133402 Hankel transform of A006588.

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A228484 a(n) = 2^n*(3*n)!/(n!*(2*n)!).

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A006587 a(n) = 3*2^(2*n)*(3*n)!/((2*n)!*n!).

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A137209 Decimal expansion of (1/2)*sqrt(3/Pi).

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A378779 a(n) = n^2 * 4^n * binomial(3*n, n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A228484 a(n) = 2^n(3n)!/(n!(2n)!).

A006587 a(n) = 32^(2n)(3n)!/((2n)!n!).