A056982 a(n) = 4^A005187(n). The denominators of the Landau constants.

1, 4, 64, 256, 16384, 65536, 1048576, 4194304, 1073741824, 4294967296, 68719476736, 274877906944, 17592186044416, 70368744177664, 1125899906842624, 4503599627370496, 4611686018427387904, 18446744073709551616, 295147905179352825856, 1180591620717411303424

Offset

0,2

Comments

Also equal to A046161(n)^2.

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.

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.

Denominator of h^(2n) in the Kummer-Gauss series for the perimeter of an ellipse.

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

Also denominators of coefficients in hypergeometric([1/2,1/2],[1],x). The numerators are given in A038534. - Wolfdieter Lang, May 29 2016

Also denominators of A277233. - Wolfdieter Lang, Nov 16 2016

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

References

J.-P. Delahaye, Pi - die Story (German translation), Birkhäuser, 1999 Basel, p. 84. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997.

O. J. Farrell and B. Ross, Solved Problems in Analysis, Dover, NY, 1971; p. 77.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..500

B. Gourevitch, L'univers de Pi

Edmund Landau, Abschätzung der Koeffzientensumme einer Potenzreihe, Arch. Math. Phys. 21 (1913), 42-50. [Accessible in the USA through the Hathi Trust Digital Library.]

Edmund Landau, Abschätzung der Koeffzientensumme einer Potenzreihe (Zweite Abhandlung), Arch. Math. Phys. 21 (1913), 250-255.  [Accessible in the USA through the Hathi Trust Digital Library.]

Cristinel Mortici, Sharp bounds of the Landau constants, Math. Comp. 80 (2011), pp. 1011-1018.

G. N. Watson, The constants of Landau and Lebesgue, Quart. J. Math. Oxford Ser. 1:2 (1930), pp. 310-318.

Eric Weisstein's World of Mathematics, Gauss-Kummer Series

Eric Weisstein's World of Mathematics, Ellipse

Index to divisibility sequences

Formula

a(n) = (denominator(binomial(1/2, n)))^2. - Peter Luschny, Sep 27 2019

Maple

A056982 := n -> denom(binomial(1/2, n))^2:
seq(A056982(n), n=0..19); # Peter Luschny, Apr 08 2016
# Alternatively:
G := proc(x) hypergeom([1/2, 1/2], [1], x)/(1-x) end: ser := series(G(x), x, 20):
[seq(coeff(ser, x, n), n=0..19)]: denom(%); # Peter Luschny, Sep 28 2019

Mathematica

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 *)
G[x_] := (2 EllipticK[x])/(Pi (1 - x));
CoefficientList[Series[G[x], {x, 0, 19}], x] // Denominator (* Peter Luschny, Sep 28 2019 *)

Prog

(PARI) a(n)=my(s=n); while(n>>=1, s+=n); 4^s \\ Charles R Greathouse IV, Apr 07 2012

Crossrefs

Apart from offset, identical to A110258.

Cf. A005187, A046161, A056981, A069955.

Equals (1/2)*A038533(n), A038534, A277233.

Cf. A001790/A046161.

Sequence in context: A056229 A062271 A110258 * A030994 A299147 A141046

Adjacent sequences: A056979 A056980 A056981 * A056983 A056984 A056985

Keyword

nonn,frac

Author

Eric W. Weisstein

Extensions

Edited by N. J. A. Sloane, Feb 18 2004, Jun 05 2007

Status

approved