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A002595 Denominators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).

%I A002595 M4233 N1768 #47 Feb 16 2025 08:32:26
%S A002595 1,6,40,112,1152,2816,13312,10240,557056,1245184,5505024,12058624,
%T A002595 104857600,226492416,973078528,2080374784,23622320128,30064771072,
%U A002595 635655159808,446676598784,11269994184704,23639499997184,6597069766656
%N A002595 Denominators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).
%C A002595 arcsin(x) is usually written as x + x^3/(2*3) + 1*3*x^5/(2*4*5) + 1*3*5*x^7/(2*4*6*7) + ..., = x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 + ... when reduced to lowest terms.
%C A002595 arccos(x) = Pi/2 - (x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 + ...).
%C A002595 arccsc(x) = 1/x+1/(6*x^3)+3/(40*x^5)+5/(112*x^7)+35/(1152*x^9)+63/(2816*x^11)+...
%C A002595 arcsec(x) = Pi/2 -(1/x+1/(6*x^3)+3/(40*x^5)+5/(112*x^7)+35/(1152*x^9)+63/(2816*x^11)+...)
%C A002595 arcsinh(x) = x-1/6*x^3+3/40*x^5-5/112*x^7+35/1152*x^9-63/2816*x^11+...
%C A002595 arccsc(x) = arcsin(1/x) and arcsec(x) = arccos(1/x): 1 < |x|
%C A002595 arccsch(x) = arcsinh(1/x) for 1 < |x|
%C A002595 Also denominator of (2n-1)!! / ((2n+1)*(2n)!!) (n=>0).
%D A002595 W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
%D A002595 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
%D A002595 H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, Chap. 3.
%D A002595 Focus, vol. 16, no. 5, page 32, Oct 1996.
%D A002595 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002595 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002595 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 31, equation 31:6:1 at page 290.
%H A002595 T. D. Noe, <a href="/A002595/b002595.txt">Table of n, a(n) for n=0..200</a>
%H A002595 H. E. Salzer, <a href="http://dx.doi.org/10.1090/S0025-5718-1948-0023123-5">Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials</a>, Math. Comp., 3 (1948), 16-18.
%H A002595 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InverseCosecant.html">Inverse Cosecant</a>.
%H A002595 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InverseCosine.html">Inverse Cosine</a>.
%H A002595 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InverseSecant.html">Inverse Secant</a>.
%H A002595 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InverseSine.html">Inverse Sine</a>.
%H A002595 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InverseHyperbolicCosecant.html">Inverse Hyperbolic Cosecant</a>.
%H A002595 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InverseHyperbolicSine.html">Inverse Hyperbolic Sine</a>.
%H A002595 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ArchimedesSpiral.html">Archimedes' Spiral</a>.
%H A002595 Herbert S. Wilf, <a href="https://www2.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, Academic Press, NY, 1994. See p. 54.
%F A002595 a(n) = denominator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))). - _Johannes W. Meijer_, Jul 06 2009
%t A002595 Denominator[Take[CoefficientList[Series[ArcSin[x],{x,0,50}],x],{2,-1,2}]] (* _Harvey P. Dale_, Aug 06 2012 *)
%o A002595 (PARI) a(n) = denominator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))); \\ _Stefano Spezia_, Dec 31 2024
%Y A002595 A055786(n) / a(n) = A001147(n) / ( A000165(n) * (2*n+1))
%Y A002595 Cf. A162443 where BG1[ -3,n] = (-1)*A002595(n-1)/A055786(n-1) for n =>1. - _Johannes W. Meijer_, Jul 06 2009
%Y A002595 a(n) = 2*A143582(n+1) for n>=1. - _Filip Zaludek_, Oct 25 2016
%K A002595 nonn,frac,nice,easy
%O A002595 0,2
%A A002595 _N. J. A. Sloane_