A038534 Numerators of coefficients of EllipticK/Pi.

1, 1, 9, 25, 1225, 3969, 53361, 184041, 41409225, 147744025, 2133423721, 7775536041, 457028729521, 1690195005625, 25145962430625, 93990019574025, 90324408810638025, 340357374376418025, 5147380044581630625, 19520119892056100625, 1187604094232693162025

Offset: 0

Author

Wouter Meeussen, revised Jan 03 2001

Keywords

• frac
• nonn

Comments

The denominators are given in A038533.

Also numerators in expansion of the hypergeometric series 2F1(1/2,1/2; 1; x).

This means numerators of the expansion coefficients of 2*K(k)/Pi = 2F1(1/2,1/2; 1; k^2) in powers of k^2, with K(k) the complete elliptic integral of the first kind. The denominators are given in A056982. The period T of the plane pendulum (mass m, length L, Earth's gravity g, energy E) is 4*sqrt(L/g)*K(sin(phi_0/2)) with cos(phi_0) = -E/(m*g*L) (maximal phi value). See the Landau - Lifschitz reference, p. 30. - Wolfdieter Lang, May 29 2016

It is easy and inexpensive to make a satisfactory precision measurement of a(1)/4, a(2)/64, and a(3)/256 using a pendulum rigged from a computer mouse. In "Digital Pendulum Data Analysis" (see links) amplitude vs. time data is transformed to period vs. sin(phi_0/2)^2 data, thus allowing extraction of expansion coefficients as fit parameters. - Bradley Klee, Dec 25 2016

References

• B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 91, Eq. 2.1.
• L. D. Landau und E. M. Lifschitz, Mechanik, Akademie Verlag, Berlin, 1967, p. 30 (Exercise 1 in chapter III, paragraph 11.)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• B. Klee, Digital Pendulum Data Analysis: Output, Github, 2016.
• David P. Roberts and Fernando Rodriguez Villegas, Hypergeometric Motives, arXiv:2109.00027 [math.AG], 2021. See (1.2) p. 1.
• G. N. Watson, A Note on Gamma Functions.Edinburgh Mathematical Notes, 42, 1959, pp 7-9.

Crossrefs

Cf. A001790, A038533, A038535, A056040, A056982, A060632.

Programs

Maple

swing := proc(n) option remember; if n = 0 then 1 elif n mod 2 = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i, i = convert(iquo(n, 2), base, 2))):
a := n -> (swing(2*n)/sigma(2*n))^2; seq(a(n),n=0..20); # Peter Luschny, Aug 06 2014

Mathematica

Numerator@ CoefficientList[ Series[ EllipticK@x, {x, 0, 19}]/Pi, x] (* Robert G. Wilson v, Jul 19 2007 *)

Formula

a(n) = 2^(-2*w(n))*binomial(2*n,n)^2 with w(n) = A000120(n), the number of 1's in binary expansion of n.

a(n) = A001790(n)^2.

a(n) = (A056040(2*n)/A060632(2*n))^2. - Peter Luschny, Aug 06 2014

a(n) = (-1)^n*A056982(n)*C(-1/2,n)*C(n-1/2,n). - Peter Luschny, Apr 08 2016

a(n) = numerator(((2*n)!/(2^(2*n)*(n!)^2))^2). - Stefano Spezia, May 01 2025