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

%I A052469 #41 Feb 16 2025 08:32:42
%S A052469 4,32,96,1024,2560,4096,28672,524288,1179648,5242880,11534336,
%T A052469 100663296,218103808,939524096,134217728,68719476736,146028888064,
%U A052469 206158430208,1305670057984,2199023255552,7696581394432,96757023244288,202310139510784,1125899906842624
%N A052469 Denominators in the Taylor series for arccosh(x) - log(2*x).
%D A052469 Bronstein-Semendjajew, sprawotchnik po matematikje, 6th Russian ed. 1956, ch. 4.2.6.
%H A052469 Vincenzo Librandi, <a href="/A052469/b052469.txt">Table of n, a(n) for n = 1..1000</a>
%H A052469 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InverseHyperbolicCosecant.html">Inverse Hyperbolic Cosecant</a>
%H A052469 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InverseHyperbolicCosine.html">Inverse Hyperbolic Cosine</a>
%H A052469 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InverseHyperbolicSine.html">Inverse Hyperbolic Sine</a>
%F A052469 A052468(n) / a(n) = A001147(n) / ( A000165(n) *2*n )
%F A052469 From _Johannes W. Meijer_, Jul 06 2009: (Start)
%F A052469 a(n) = denom((2*n-1)!/( 4^n * (n!)^2)).
%F A052469 Equals 2*A162442(n+1) for n >= 1.
%F A052469 A052468(n)/a(n) = (1/(2*n))*A001790(n)/A046161(n) for n>=1.
%F A052469 (End)
%e A052469 arccosh(x) = log(2x) - 1/(4*x^2) - 3/(32*x^4) - 5/(96*x^6) - ... for x>1.
%t A052469 a[n_] := Denominator[(2*n-1)!/(2^(2*n)*n!^2)]; Array[a, 21] (* _Jean-François Alcover_, May 17 2017 *)
%o A052469 (Magma) [Denominator(Factorial(2*n-1)/( 2^(2*n)* Factorial(n)^2)): n in [1..30]]; // _Vincenzo Librandi_, Jul 10 2017
%o A052469 (PARI) {a(n) = denominator((2*n-1)!/(4^n*(n!)^2))}; \\ _G. C. Greubel_, May 18 2019
%o A052469 (Sage) [denominator(factorial(2*n-1)/(4^n*(factorial(n))^2)) for n in (1..30)] # _G. C. Greubel_, May 18 2019
%o A052469 (GAP) List([1..30], n-> DenominatorRat( Factorial(2*n-1)/(4^n*(Factorial(n))^2) )) # _G. C. Greubel_, May 18 2019
%Y A052469 Cf. A002595.
%K A052469 nonn,easy,frac
%O A052469 1,1
%A A052469 _Eric W. Weisstein_
%E A052469 Updated by _Frank Ellermann_, May 22 2001