A277233 -id:A277233 - 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

A327495 a(n) = numerator( Sum_{j=0..n} (j!/(2^j*floor(j/2)!)^2)^2 ).

Original entry on oeis.org

1, 17, 69, 1113, 17817, 285297, 1141213, 18260633, 1168681737, 18699007017, 74796032037, 1196736992841, 19147791938817, 306364680039081, 1225458720340365, 19607339566855065, 5019478929156305865, 80311662878468159865, 321246651514020383485, 5139946424277661728785

Offset: 0

Peter Luschny, Sep 27 2019

This sequence is a variant of the Landau constants when the normalized central binomial is replaced by the normalized swinging factorial.
(1) A277233(n)/4^A005187(n) are the Landau constants. These constants are defined as G(n) = Sum_{j=0..n} g(j)^2 with the normalized central binomial
    g(n) = (2*n)! / (2^n*n!)^2 = A001790(n)/A046161(n).
(2) A327495(n)/4^A327492(n) are the rationals considered here. These numbers are defined as H(n) = Sum_{j=0..n} h(j)^2 with the normalized swinging factorial
    h(n) = n! / (2^n*floor(n/2)!)^2 = A163590(n)/A327493(n).
(3) In particular, this means that we have the pure integer representations
    A277233(n) = Sum_{k=0..n}(A001790(k)*(2^(A005187(n) - A005187(k))))^2;
    A327495(n) = Sum_{k=0..n}(A163590(k)*(2^(A327492(n) - A327492(k))))^2.
(4) A163590 is the odd part of the swinging factorial and A001790 is the odd part of the swinging factorial at even indices (see a comment from Aug 01 2009 in A001790). Similarly, A327493(2n)=A046161(2n) and A327493(2n+1) = 2*A046161(2n+1).
(5) A005187 are the partial sums of A001511, the 2-adic valuation of 2n, and A327492 are the partial sums of A327491.

			r(n) = 1, 17/16, 69/64, 1113/1024, 17817/16384, 285297/262144, 1141213/1048576, 18260633/16777216, ...

Denominators in A327496.
Cf. A327491, A327492, A327493, A327494.
Cf. A277233, A005187, A056040, A000984, A001790/A046161, A163590, A001511.

A327495 := n -> numer(add(j!^2/(2^j*iquo(j,2)!)^4, j=0..n)):
seq(A327495(n), n=0..19);

a(n)={ numerator(sum(j=0, n, (j!/(2^j*(j\2)!)^2)^2 )) } \\ Andrew Howroyd, Sep 28 2019

Denominator(r(n)) = 4^A327492(n) = A327493(n)^2 = A327496(n).

a(n) = Sum_{k=0..n} (A163590(k)*(2^(A327492(n) - A327492(k))))^2.

A212298 Decimal expansion of (gamma + log(16))/Pi.

Original entry on oeis.org

1, 0, 6, 6, 2, 7, 5, 8, 5, 3, 2, 0, 8, 9, 1, 4, 3, 5, 4, 3, 4, 5, 1, 1, 0, 1, 9, 6, 6, 1, 5, 7, 4, 6, 9, 4, 6, 7, 5, 8, 0, 1, 7, 5, 5, 6, 0, 3, 9, 9, 0, 4, 3, 0, 6, 6, 7, 9, 2, 2, 7, 3, 5, 1, 5, 7, 7, 6, 1, 2, 7, 0, 6, 8, 8, 5, 9, 8, 2, 8, 7, 1, 2, 5, 0, 4, 4

Offset: 1

Charles R Greathouse IV, Jul 02 2012

Decimal expansion of c, where G_n = A277233(n)/A056982(n) = c + log(n+1)/Pi - 1/(4 Pi(n+1)) + O(1/n^2). [Corrected by Peter Luschny, Sep 27 2019]

Decimal expansion of c, where G_n = A277233(n)/A056982(n) = c + log(n+3/4)/Pi + O(1/n^2). - Peter Luschny, Sep 27 2019

			1.06627585320891435434511019661574694675801755603990430667922735157761...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 4.2.1, p. 252.

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

Cf. A277233/A056982.

Digits := 120: ((gamma + 4*log(2))/Pi)*10^94:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Sep 27 2019

RealDigits[(EulerGamma + Log[16])/Pi, 10, 87][[1]] (* Georg Fischer, Apr 04 2020 *)

PARI
```
(Euler+log(16))/Pi
```

Definition amended by Georg Fischer, Apr 04 2020

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

A327495 a(n) = numerator( Sum_{j=0..n} (j!/(2^j*floor(j/2)!)^2)^2 ).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Formula

A212298 Decimal expansion of (gamma + log(16))/Pi.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions