|
%I #48 Sep 28 2019 17:57:45
%S 1,4,64,256,16384,65536,1048576,4194304,1073741824,4294967296,
%T 68719476736,274877906944,17592186044416,70368744177664,
%U 1125899906842624,4503599627370496,4611686018427387904,18446744073709551616,295147905179352825856,1180591620717411303424
%N a(n) = 4^A005187(n). The denominators of the Landau constants.
%C Also equal to A046161(n)^2.
%C Let W(n) = Product_{k=1..n} (1- 1/(4*k^2)), the partial Wallis product with lim n -> infinity W(n) = 2/Pi; a(n) = denominator(W(n)). The numerators are in A069955.
%C Equivalently, denominators in partial products of the following approximation to Pi: Pi = Product_{n >= 1} 4*n^2/(4*n^2-1). Numerators are in A069955.
%C Denominator of h^(2n) in the Kummer-Gauss series for the perimeter of an ellipse.
%C Denominators of coefficients in hypergeometric([1/2,-1/2],[1],x). The numerators are given in A038535. hypergeom([1/2,-1/2],[1],e^2) = L/(2*Pi*a) with the perimeter L of an ellipse with major axis a and numerical eccentricity e (Maclaurin 1742). - _Wolfdieter Lang_, Nov 08 2010
%C Also denominators of coefficients in hypergeometric([1/2,1/2],[1],x). The numerators are given in A038534. - _Wolfdieter Lang_, May 29 2016
%C Also denominators of A277233. - _Wolfdieter Lang_, Nov 16 2016
%C A277233(n)/a(n) are the Landau constants. These constants are defined as G(n) = Sum_{j=0..n} g(j)^2, where g(n) = (2*n)!/(2^n*n!)^2 = A001790(n)/A046161(n). - _Peter Luschny_, Sep 27 2019
%D J.-P. Delahaye, Pi - die Story (German translation), Birkhäuser, 1999 Basel, p. 84. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997.
%D O. J. Farrell and B. Ross, Solved Problems in Analysis, Dover, NY, 1971; p. 77.
%H Charles R Greathouse IV, <a href="/A056982/b056982.txt">Table of n, a(n) for n = 0..500</a>
%H B. Gourevitch, <a href="http://www.pi314.net">L'univers de Pi</a>
%H Edmund Landau, <a href="https://babel.hathitrust.org/cgi/pt?id=uc1.a0002839165&view=1up&seq=56">Abschätzung der Koeffzientensumme einer Potenzreihe</a>, Arch. Math. Phys. 21 (1913), 42-50. [Accessible in the USA through the <a href="https://www.hathitrust.org/accessibility">Hathi Trust Digital Library</a>.]
%H Edmund Landau, <a href="https://babel.hathitrust.org/cgi/pt?id=uc1.a0002839165&view=1up&seq=264">Abschätzung der Koeffzientensumme einer Potenzreihe (Zweite Abhandlung)</a>, Arch. Math. Phys. 21 (1913), 250-255. [Accessible in the USA through the <a href="https://www.hathitrust.org/accessibility">Hathi Trust Digital Library</a>.]
%H Cristinel Mortici, <a href="http://dx.doi.org/10.1090/S0025-5718-2010-02428-7">Sharp bounds of the Landau constants</a>, Math. Comp. 80 (2011), pp. 1011-1018.
%H G. N. Watson, <a href="https://doi.org/10.1093/qmath/os-1.1.310">The constants of Landau and Lebesgue</a>, Quart. J. Math. Oxford Ser. 1:2 (1930), pp. 310-318.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Gauss-KummerSeries.html">Gauss-Kummer Series</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Ellipse.html">Ellipse</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F a(n) = (denominator(binomial(1/2, n)))^2. - _Peter Luschny_, Sep 27 2019
%p A056982 := n -> denom(binomial(1/2, n))^2:
%p seq(A056982(n), n=0..19); # _Peter Luschny_, Apr 08 2016
%p # Alternatively:
%p G := proc(x) hypergeom([1/2,1/2], [1], x)/(1-x) end: ser := series(G(x), x, 20):
%p [seq(coeff(ser,x,n), n=0..19)]: denom(%); # _Peter Luschny_, Sep 28 2019
%t Table[Power[4, 2 n - DigitCount[2 n, 2, 1]], {n, 0, 19}] (* _Michael De Vlieger_, May 30 2016, after _Harvey P. Dale_ at A005187 *)
%t G[x_] := (2 EllipticK[x])/(Pi (1 - x));
%t CoefficientList[Series[G[x], {x, 0, 19}], x] // Denominator (* _Peter Luschny_, Sep 28 2019 *)
%o (PARI) a(n)=my(s=n); while(n>>=1, s+=n); 4^s \\ _Charles R Greathouse IV_, Apr 07 2012
%Y Apart from offset, identical to A110258.
%Y Cf. A005187, A046161, A056981, A069955.
%Y Equals (1/2)*A038533(n), A038534, A277233.
%Y Cf. A001790/A046161.
%K nonn,frac
%O 0,2
%A _Eric W. Weisstein_
%E Edited by _N. J. A. Sloane_, Feb 18 2004, Jun 05 2007
| 