A113424 a(n) = (6*n)!/((3*n)!*(2*n)!*n!).

1, 60, 13860, 4084080, 1338557220, 465817912560, 168470811709200, 62588625639883200, 23717177328413240100, 9124964373613212524400, 3553261127084984957001360, 1397224499394244497967972800, 553883078634868423069470550800, 221068174083308549543680044926400

Offset: 0

Author

Michael Somos, Oct 31 2005

Keywords

• nonn
• easy

Comments

Appears in Ramanujan's theory of elliptic functions of signature 6.

The family of elliptic curves "x=2*H=p^2+q^2-q^3, 0Bradley Klee, Feb 25 2018

The power series with coefficients a(n) * n! plays a central role in the Faber-Zagier relations on the moduli space of algebraic curves; see Pandharipande and Pixton, Section 0.2. - Harry Richman, Aug 19 2024

Examples

			G.f. = 1 + 60*x + 13860*x^2 + 4084080*x^3 + 1338557220*x^4 + ... - _Michael Somos_, Dec 02 2018
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• J. Cremona, Elliptic Curves over Q, LMFDB 2017.
• Alin Bostan, Armin Straub, and Sergey Yurkevich, On the representability of sequences as constant terms, arXiv:2212.10116 [math.NT], 2022.
• H. J. Brothers, Pascal's Prism: Supplementary Material.
• S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 23.
• Bradley Klee, Geometric G.F. for Ramanujan Periods, seqfans mailing list, 2017.
• Bradley Klee, On LMFDB period data, LMFDB-support mailing list, 2018.
• Bradley Klee, Weierstrass Solution of Cubic Anharmonic Oscillation, Wolfram Demonstrations Project, 2018.
• Romeo Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
• R. Pandharipande and A. Pixton, Relations in the tautological ring of the moduli space of curves, arXiv:1301.4561 [math.AG], 2020.
• S. Ramanujan, Modular Equations and Approximations to Pi, Quarterly Journal of Mathematics, XLV (1914), 350-372.
• L. C. Shen, A note on Ramanujan's identities involving the hypergeometric function 2F1(1/6,5/6;1;z), The Ramanujan Journal, 30.2 (2013), 211-222.

Crossrefs

a(n) = A347304(6*n)

Elliptic Integrals: A002894, A006480, A000897. Factors: A005809, A066802.

Cf. A188662.

Programs

GAP

List([0..15],n->Factorial(6*n)/(Factorial(3*n)*Factorial(2*n)*Factorial(n))); # Muniru A Asiru, Apr 08 2018

Mathematica

a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/6, 5/6, 1, 432 x], {x, 0, n}];
Table[Multinomial[n, 2 n, 3 n], {n, 0, 15}] (* Vladimir Reshetnikov, Oct 12 2016 *)
a[ n_] := Multinomial[n, 2 n, 3 n]; (* Michael Somos, Dec 02 2018 *)

PARI

{a(n) = if( n<0, 0, (6*n)! / ((3*n)! * (2*n)! * n!))};

Formula

G.f.: hypergeometric2F1(1/6, 5/6; 1; 432 * x).

a(n) ~ 432^n/(2*Pi*n). - Ilya Gutkovskiy, Oct 13 2016

a(n) = A005809(n)*A066802(n). - Bradley Klee, Feb 25 2018

0 = a(n)*(-267483013447680*a(n+2) [phone]*a(n+3) -204669037440*a(n+4) [phone]*a(n+5)) +a(n+1)*([phone]*a(n+2) -57870650880*a(n+3) [phone]*a(n+4) -1088188*a(n+5)) +a(n+2)*(-17884800*a(n+2) [phone]*a(n+3) - 295844*a(n+4) +693*a(n+5)) for all n in Z. - Michael Somos, May 16 2018

From Peter Bala, Feb 28 2020: (Start)

a(n) = C(6*n,2*n)*C(4*n,n).

a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k (apply Mestrovic, equation 39).

(-1)^n*a(n) = [x^(2*n)*y^(2*n)] ( (1 + x + y)*(1 - x + y) )^(4*n).

a(n) = [x^n] ( F(x) )^(60*n), where F(x) = 1 + x + 56*x^2 + 7355*x^3 + 1290319*x^4 + 264117464*x^5 + 59508459679*x^6 + ... appears to have integer coefficients. We conjecture that for k >= 1 the sequence defined by b_k(n) := [x^n] F(x)^(k*n) satisfies the above supercongruences for primes p >= 7. (End)

From Peter Bala, Mar 20 2022: (Start)

Right-hand side of the following identities valid for n >= 1:

Sum_{k = 0..2*n} 4*n*(4*n+k-1)!/(k!*n!*(3*n)!) = (6*n)!/((3*n)!*(2*n)!*n!);

Sum_{k = 0..3*n} 3*n*(3*n+k-1)!/(k!*n!*(2*n)!) = (6*n)!/((3*n)!(2*n)!*n!).

Cf. A001451. (End)

From Peter Bala, Feb 26 2023: (Start)

a(n) = (4^n/n!^2) * Product_{k = n..3*n-1} 2*k + 1.

a(n) = (12^n/n!^2) * Product_{k = 0..n-1} (6*k + 1)*(6*k + 5). (End)

a(n) = 12*(6*n - 1)*(6*n - 5)*a(n-1)/n^2. - Neven Sajko, Jul 19 2023

From Karol A. Penson, Dec 26 2023: (Start)

a(n) = Integral_{x=0..432} x^n*W(x) dx, n>=0, where W(x) = sqrt(18)*MeijerG([[], [0, 0]], [[-1/6, -5/6], []], x/432)/(1296*Pi), where MeijerG is the Meijer G - function.

Apparently, W(x) cannot be represented by any other function. W(x) is positive on x = [0, 432], it diverges at x=0, and monotonically decreases for x>0. It appears that at x=432, W(x) tends to a constant value close to 0.000368414. This integral representation as the n-th power moment of the positive function W(x) on the interval [0, 432] is unique, as W(x) is the solution of the Hausdorff moment problem. (End)

W(x) can be represented in terms of two 2F1 hypergeometric functions, W(x) = hypergeom([1/6, 1/6], [1/3], x/432)/(6*sqrt(Pi)*Gamma(2/3)*Gamma(5/6)*x^(5/6)) - Gamma(2/3)*Gamma(5/6)*sqrt(3)*hypergeom([5/6, 5/6], [5/3], x/432)/(1152*Pi^(5/2)*x^(1/6)), x on (0, 432). - Karol A. Penson, May 16 2025