This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A038534 #47 May 02 2025 03:17:29
%S A038534 1,1,9,25,1225,3969,53361,184041,41409225,147744025,2133423721,
%T A038534 7775536041,457028729521,1690195005625,25145962430625,93990019574025,
%U A038534 90324408810638025,340357374376418025,5147380044581630625,19520119892056100625,1187604094232693162025
%N A038534 Numerators of coefficients of EllipticK/Pi.
%C A038534 The denominators are given in A038533.
%C A038534 Also numerators in expansion of the hypergeometric series 2F1(1/2,1/2; 1; x).
%C A038534 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
%C A038534 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
%D A038534 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 91, Eq. 2.1.
%D A038534 L. D. Landau und E. M. Lifschitz, Mechanik, Akademie Verlag, Berlin, 1967, p. 30 (Exercise 1 in chapter III, paragraph 11.)
%H A038534 Vincenzo Librandi, <a href="/A038534/b038534.txt">Table of n, a(n) for n = 0..200</a>
%H A038534 B. Klee, <a href="https://github.com/bradklee/DataAnalysis/tree/master/PlanePendulum/Output">Digital Pendulum Data Analysis: Output</a>, Github, 2016.
%H A038534 David P. Roberts and Fernando Rodriguez Villegas, <a href="https://arxiv.org/abs/2109.00027">Hypergeometric Motives</a>, arXiv:2109.00027 [math.AG], 2021. See (1.2) p. 1.
%H A038534 G. N. Watson, <a href="http://dx.doi.org/10.1017/S0950184300003207">A Note on Gamma Functions.</a>Edinburgh Mathematical Notes, 42, 1959, pp 7-9.
%F A038534 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.
%F A038534 a(n) = A001790(n)^2.
%F A038534 a(n) = (A056040(2*n)/A060632(2*n))^2. - _Peter Luschny_, Aug 06 2014
%F A038534 a(n) = (-1)^n*A056982(n)*C(-1/2,n)*C(n-1/2,n). - _Peter Luschny_, Apr 08 2016
%F A038534 a(n) = numerator(((2*n)!/(2^(2*n)*(n!)^2))^2). - _Stefano Spezia_, May 01 2025
%p A038534 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:
%p A038534 sigma := n -> 2^(add(i, i = convert(iquo(n, 2), base, 2))):
%p A038534 a := n -> (swing(2*n)/sigma(2*n))^2; seq(a(n),n=0..20); # _Peter Luschny_, Aug 06 2014
%t A038534 Numerator@ CoefficientList[ Series[ EllipticK@x, {x, 0, 19}]/Pi, x] (* _Robert G. Wilson v_, Jul 19 2007 *)
%Y A038534 Cf. A001790, A038533, A038535, A056040, A056982, A060632.
%K A038534 frac,nonn
%O A038534 0,3
%A A038534 _Wouter Meeussen_, revised Jan 03 2001