A228484 -id:A228484 - cp's OEIS frontend

A378780 a(n) = n * 2^n * binomial(3*n, n).

0, 6, 120, 2016, 31680, 480480, 7128576, 104186880, 1506244608, 21596889600, 307660953600, 4359995228160, 61522462310400, 865005820084224, 12124867905454080, 169509237023047680, 2364380454476316672, 32913250644698726400, 457355892992216924160, 6345297974846973542400

Offset: 0

Amiram Eldar, Dec 07 2024

Jonathan Borwein, David Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Natick, MA, 2004. See p. 26.

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. See p. 379, eq. (3.9).
Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36. See p. 32, eq. (43).

Cf. A005809, A036289, A228484.

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

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

a(n) = A036289(n) * A005809(n).
a(n) = n * A228484(n).
a(n) == 0 (mod 6).
Sum_{n>=1} 1/a(n) = Pi/10 - log(2)/5 (Borwein et al., 2004; Borwein and Girgensohn, 2005; Batir, 2005).

A386918 a(n) = 2^n * binomial(4*n,n).

Original entry on oeis.org

1, 8, 112, 1760, 29120, 496128, 8614144, 151557120, 2692684800, 48201359360, 868004380672, 15706806542336, 285362317180928, 5202031080243200, 95104728494899200, 1743063914667048960, 32016101348447354880, 589188508080622534656, 10861173739509105295360

Offset: 0

Seiichi Manyama, Aug 08 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..350

Cf. A066381, A386919, A386920.

Cf. A005810, A059304, A228484, A378804.

[2^n * Binomial(4*n,n): n in [0..26]]; // Vincenzo Librandi, Aug 11 2025

Table[2^n*Binomial[4*n,n],{n,0,30}] (* Vincenzo Librandi, Aug 11 2025 *)

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

a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(4*n-k,n-k).
a(n) = [x^n] (1+x)^(4*n)/(1-x)^(3*n+1).
a(n) = [x^n] 1/(1-2*x)^(3*n+1).
a(n) = [x^n] (1+2*x)^(4*n).

A378778 a(n) = n^2 * 2^n * binomial(3*n, n).

Original entry on oeis.org

0, 6, 240, 6048, 126720, 2402400, 42771456, 729308160, 12049956864, 194372006400, 3076609536000, 47959947509760, 738269547724800, 11245075661094912, 169748150676357120, 2542638555345715200, 37830087271621066752, 559525260959878348800, 8232406073859904634880, 120560661522092497305600

Offset: 0

Amiram Eldar, Dec 07 2024

Jonathan Borwein, David Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Natick, MA, 2004. See p. 26.

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. See p. 379, eq. (3.6).
Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36. See p. 32, eq. (43).

Cf. A005809, A007758, A228484, A378780.

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

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

a(n) = A007758(n) * A005809(n).
a(n) = n^2 * A228484(n).
a(n) = n * A378780(n).
a(n) == 0 (mod 6).
Sum_{n>=1} 1/a(n) = Pi^2/24 - log(2)^2/2 (Borwein et al., 2004; Borwein and Girgensohn, 2005; Batir, 2005).

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

cp's OEIS Frontend

A378780 a(n) = n * 2^n * binomial(3*n, n).

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A386918 a(n) = 2^n * binomial(4*n,n).

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A378778 a(n) = n^2 * 2^n * binomial(3*n, n).

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

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