A052469 Denominators in the Taylor series for arccosh(x) - log(2*x).

4, 32, 96, 1024, 2560, 4096, 28672, 524288, 1179648, 5242880, 11534336, 100663296, 218103808, 939524096, 134217728, 68719476736, 146028888064, 206158430208, 1305670057984, 2199023255552, 7696581394432, 96757023244288, 202310139510784, 1125899906842624

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy
• frac

Examples

			arccosh(x) = log(2x) - 1/(4*x^2) - 3/(32*x^4) - 5/(96*x^6) - ... for x>1.
		

References

• Bronstein-Semendjajew, sprawotchnik po matematikje, 6th Russian ed. 1956, ch. 4.2.6.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosecant
• Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosine
• Eric Weisstein's World of Mathematics, Inverse Hyperbolic Sine

Crossrefs

Cf. A002595.

Programs

GAP

List([1..30], n-> DenominatorRat( Factorial(2*n-1)/(4^n*(Factorial(n))^2) )) # G. C. Greubel, May 18 2019

Magma

[Denominator(Factorial(2*n-1)/( 2^(2*n)* Factorial(n)^2)): n in [1..30]]; // Vincenzo Librandi, Jul 10 2017

Mathematica

a[n_] := Denominator[(2*n-1)!/(2^(2*n)*n!^2)]; Array[a, 21] (* Jean-François Alcover, May 17 2017 *)

PARI

{a(n) = denominator((2*n-1)!/(4^n*(n!)^2))}; \\ G. C. Greubel, May 18 2019

Sage

[denominator(factorial(2*n-1)/(4^n*(factorial(n))^2)) for n in (1..30)] # G. C. Greubel, May 18 2019

Formula

A052468(n) / a(n) = A001147(n) / ( A000165(n) *2*n )

From Johannes W. Meijer, Jul 06 2009: (Start)

a(n) = denom((2*n-1)!/( 4^n * (n!)^2)).

Equals 2*A162442(n+1) for n >= 1.

A052468(n)/a(n) = (1/(2*n))*A001790(n)/A046161(n) for n>=1.

(End)

Extensions

Updated by Frank Ellermann, May 22 2001