A002595 - OEIS

%I M4233 N1768 #46 Feb 10 2025 03:14:18

%S 1,6,40,112,1152,2816,13312,10240,557056,1245184,5505024,12058624,

%T 104857600,226492416,973078528,2080374784,23622320128,30064771072,

%U 635655159808,446676598784,11269994184704,23639499997184,6597069766656

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

%C 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 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 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 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 arcsinh(x) = x-1/6*x^3+3/40*x^5-5/112*x^7+35/1152*x^9-63/2816*x^11+...

%C arccsc(x) = arcsin(1/x) and arcsec(x) = arccos(1/x): 1 < |x|

%C arccsch(x) = arcsinh(1/x) for 1 < |x|

%C Also denominator of (2n-1)!! / ((2n+1)*(2n)!!) (n=>0).

%D 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 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.

%D H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, Chap. 3.

%D Focus, vol. 16, no. 5, page 32, Oct 1996.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 31, equation 31:6:1 at page 290.

%H T. D. Noe, <a href="/A002595/b002595.txt">Table of n, a(n) for n=0..200</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseCosecant.html">Inverse Cosecant</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseCosine.html">Inverse Cosine</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseSecant.html">Inverse Secant</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseSine.html">Inverse Sine</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseHyperbolicCosecant.html">Inverse Hyperbolic Cosecant</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseHyperbolicSine.html">Inverse Hyperbolic Sine</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ArchimedesSpiral.html">Archimedes' Spiral</a>.

%H Herbert S. Wilf, <a href="https://www2.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, Academic Press, NY, 1994. See p. 54.

%F a(n) = denominator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))). - _Johannes W. Meijer_, Jul 06 2009

%t Denominator[Take[CoefficientList[Series[ArcSin[x],{x,0,50}],x],{2,-1,2}]] (* _Harvey P. Dale_, Aug 06 2012 *)

%o (PARI) a(n) = denominator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))); \\ _Stefano Spezia_, Dec 31 2024

%Y A055786(n) / a(n) = A001147(n) / ( A000165(n) * (2*n+1))

%Y Cf. A162443 where BG1[ -3,n] = (-1)*A002595(n-1)/A055786(n-1) for n =>1. - _Johannes W. Meijer_, Jul 06 2009

%Y a(n) = 2*A143582(n+1) for n>=1. - _Filip Zaludek_, Oct 25 2016

%K nonn,frac,nice,easy,changed

%O 0,2

%A _N. J. A. Sloane_