A069955 -id:A069955 - cp's OEIS frontend

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

Eric W. Weisstein

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

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.

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

Apart from offset, identical to A110258.
Cf. A005187, A046161, A056981, A069955.
Equals (1/2)*A038533(n), A038534, A277233.
Cf. A001790/A046161.

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

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 *)

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

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

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

A124399 a(n) = 4^(n - bitcount(n)) where bitcount(n) = A000120(n).

Original entry on oeis.org

1, 1, 4, 4, 64, 64, 256, 256, 16384, 16384, 65536, 65536, 1048576, 1048576, 4194304, 4194304, 1073741824, 1073741824, 4294967296, 4294967296, 68719476736, 68719476736, 274877906944, 274877906944, 17592186044416, 17592186044416

Offset: 0

Wolfdieter Lang, Nov 10 2006

Numerators of one half of norm square of monic Legendre polynomials p_n(x).
The denominators of these polynomials are given by A069955.
The rationals N2(n) = 2*a(n)/A069955(n) give the minimal norm square for real monic polynomials. The norm square is defined as integral over the interval [-1,+1] of the square of the polynomials. Cf. the Courant-Hilbert reference.

			Rationals a(n)/A069955(n): [1, 1/3, 4/45, 4/175, 64/11025, 64/43659, 256/693693, ...].
Rationals N2(n): [2, 2/3, 8/45, 8/175, 128/11025, 128/43659, 512/693693,...].

Richard Courant and David Hilbert, Methoden der mathematischen Physik, Bd. I, 3, Auflage, Springer, 1993, pp. 73-74.

Wolfdieter Lang, Norm square, rationals and more.

Cf. A000120, A001790, A056982, A060818, A069955 (denominators of N2(n) as defined in the comments).

bitcount(n) = sum(digits(n, base=2))
a(n) = 4^(n - bitcount(n)) # Peter Luschny, Oct 01 2019

a[n_] := 4^(n - DigitCount[n, 2, 1]); Array[a, 25, 0] (* Amiram Eldar, Jul 25 2023 *)

a(n) = numerator((1/(2*n+1))*((2^n)/binomial(2*n,n))^2); \\ Michel Marcus, Aug 11 2019

a(n) = numerator(N2(n)/2) with N2(n)/2:=(1/(2*n+1))*((2^n)/binomial(2*n,n))^2.
N2(n)/2 = (1/(2*n+1))*(1/L(n))^2 with L(n)= A001790(n)/A060818(n), the leading coefficient of the Legendre polynomial P_n(x), in lowest terms.
Bisection: a(2*n)=a(2*n+1) = A056982(n), n>=0.

New name by Peter Luschny, Oct 01 2019

A380949 a(n) = numerator(r(n)) where r(n) = (n/2)(Pi/2)^cos(Pi(n-1))*((n/2-1/2)!/(n/2)!)^2.

Original entry on oeis.org

0, 1, 1, 4, 9, 64, 75, 256, 1225, 16384, 19845, 65536, 160083, 1048576, 1288287, 4194304, 41409225, 1073741824, 1329696225, 4294967296, 10667118605, 68719476736, 85530896451, 274877906944, 1371086188563, 17592186044416, 21972535073125, 70368744177664, 176021737014375

Offset: 0

Peter Luschny, Feb 11 2025

			r(n) = 0, 1, 1/2, 4/3, 9/16, 64/45, 75/128, 256/175, 1225/2048, ...

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A380950 (denominator), A380910, A380909, A019267 (asymptotic coefficients).

Cf. A161736, A056982, A124399, A161737, A069955, A038534, A278145, A001901.

r := n -> (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2:
a := n -> numer(simplify(r(n))): seq(a(n), n = 0..28);
# Alternative:
r := n -> ifelse(n <= 1, n, (n - 1)/(n*r(n - 1))):

Join[{0}, Numerator[FoldList[(#2 - 1)/(#2*#) &, Range[30]]]] (* Paolo Xausa, Feb 14 2025 *)

Product_{k=1..n} a(k) = A380910(n) / A380909(n).
r(n) = (n - 1)/(n*r(n - 1)) for n > 1.
numerator(r(2*n)) = A161736(n).
numerator(r(2*n+1)) = A056982(n).
numerator(r(2*n+1))/4^n = A124399(n).
denominator(r(2*n-2)) = A161737(n).
denominator(r(2*n+1)) = A069955(n).
denominator(r(2*n+1))/(2*n+1) = A038534(n).
denominator(r(2*n+2))/2 = A278145(n).
denominator(r(2*n+2))/2^(2*n+1) = A001901(n).
r(n) ~ (2/Pi)^((-1)^n)*(1 - 1/(2*n) + 1/(8*n^2) + 1/(16*n^3) - 5/(128*n^4) - 23/(256*n^5) ...).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A124399 a(n) = 4^(n - bitcount(n)) where bitcount(n) = A000120(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Julia

Mathematica

PARI

Formula

Extensions

A380949 a(n) = numerator(r(n)) where r(n) = (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A380949 a(n) = numerator(r(n)) where r(n) = (n/2)(Pi/2)^cos(Pi(n-1))*((n/2-1/2)!/(n/2)!)^2.