Search: id:A002595
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%I A002595 M4233 N1768
%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 arcsch(x) = arsinh(1/x) for 1 < |x|
%C A002595 Also denominator of (2n-1)!! / ((2n+1)*(2n)!!) (n=>0).
%D A002595 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002595 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%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 H. E. Salzer, Coefficients for expressing the first twenty-four powers 
               in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.
%H A002595 T. D. Noe, <a href="b002595.txt">Table of n, a(n) for n=0..200</a>
%H A002595 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseCosecant.html">Inverse Cosecant</a>
%H A002595 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseCosine.html">Inverse Cosine</a>
%H A002595 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseSecant.html">Inverse Secant</a>
%H A002595 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseSine.html">Inverse Sine</a>
%H A002595 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseHyperbolicCosecant.html">Inverse Hyperbolic Cosecant</a>
%H A002595 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseHyperbolicSine.html">Inverse Hyperbolic Sine</a>
%H A002595 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ArchimedesSpiral.html">Archimedes' Spiral</a>
%F A002595 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 
               2009: (Start)
%F A002595 a(n) =denom((2*n)!/(2^(2*n)*(n)!^2*(2*n+1)))
%F A002595 (End)
%Y A002595 A055786(n) / a(n) = A001147(n) / ( A000165(n) * (2*n+1))
%Y A002595 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 
               2009: (Start)
%Y A002595 Cf. A162443 where BG1[ -3,n] = (-1)*A002595(n-1)/A055786(n-1) for n =>
               1.
%Y A002595 (End)
%Y A002595 Sequence in context: A045565 A110424 A114079 this_sequence A089207 A027777 
               A073773
%Y A002595 Adjacent sequences: A002592 A002593 A002594 this_sequence A002596 A002597 
               A002598
%K A002595 nonn,frac,nice,easy
%O A002595 0,2
%A A002595 N. J. A. Sloane (njas(AT)research.att.com).

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