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%I A055786
%S A055786 1,1,3,5,35,63,231,143,6435,12155,46189,88179,676039,1300075,5014575,
%T A055786 9694845,100180065,116680311,2268783825,1472719325,34461632205,
%U A055786 67282234305,17534158031,514589420475,8061900920775,5267108601573
%N A055786 Numerators of Taylor series expansion of arcsin(x). Also arises from 
               arccos(x), arccsc(x), arcsec(x), arcsinh(x).
%C A055786 Note that the sequence is not monotonic.
%D A055786 Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, 
               ch. 4.2.6
%D A055786 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
%D A055786 H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, 
               NY, 1968, Chap. 3.
%H A055786 T. D. Noe, <a href="b055786.txt">Table of n, a(n) for n=0..200</a>
%H A055786 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseCosecant.html">Inverse Cosecant</a>
%H A055786 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseCosine.html">Inverse Cosine</a>
%H A055786 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseSecant.html">Inverse Secant</a>
%H A055786 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseSine.html">Inverse Sine</a>
%H A055786 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseHyperbolicCosecant.html">Inverse Hyperbolic Cosecant</a>
%H A055786 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseHyperbolicCosine.html">Inverse Hyperbolic Cosine</a>
%H A055786 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               InverseHyperbolicSine.html">Inverse Hyperbolic Sine</a>
%F A055786 a(n) / A052469(n) = A001147(n) / ( A000165(n) *2*n ). E.g. a(6) = 77 
               = 1*3*5*7*9*11 / gcd( 1*3*5*7*9*11, 2*4*6*8*10*12*12 )
%F A055786 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 
               2009: (Start)
%F A055786 a(n) = numer((2*n)!/(2^(2*n)*(n)!^2*(2*n+1)))
%F A055786 (End)
%e A055786 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) + ..., which is x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/
               1152*x^9 + 63/2816*x^11 + ... (A055786/A002595) when reduced to lowest 
               terms.
%e A055786 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 + ...) (A055786/A002595).
%e A055786 arccsc(x) = 1/x+1/(6*x^3)+3/(40*x^5)+5/(112*x^7)+35/(1152*x^9)+63/(2816*x^11)+... 
               (A055786/A002595).
%e A055786 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)+...) (A055786/A002595).
%e A055786 arcsinh(x) = x-1/6*x^3+3/40*x^5-5/112*x^7+35/1152*x^9-63/2816*x^11+... 
               (A055786/A002595).
%e A055786 I*Pi/2 - arccosh(x) = I*x + 1/6*I*x^3 + 3/40*I*x^5 + 5/112*I*x^7 + 35/
               1152*I*x^9 + 63/2816*I*x^11 + 231/13312*I*x^13 + 143/10240*I*x^15 
               + 6435/557056*I*x^17 + ... (A055786/A002595).
%e A055786 0, 1, 0, 1/6, 0, 3/40, 0, 5/112, 0, 35/1152, 0, 63/2816, 0, 231/13312, 
               0, 143/10240, 0, 6435/557056, 0, 12155/1245184, 0, 46189/5505024, 
               0, ... = A055786/A002595.
%e A055786 a(4) = 35 = 3*5*7*9 / gcd( 3*5*7*9, (2*4*6*8) * (2*4+1))
%Y A055786 Cf. A002595.
%Y A055786 a(n) / A002595(n) = A001147(n) / ( A000165(n) * (2*n+1))
%Y A055786 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 
               2009: (Start)
%Y A055786 Cf. A162443 where BG1[ -3,n] = (-1)*A002595(n-1)/A055786(n-1) for n =>
               1.
%Y A055786 (End)
%Y A055786 Sequence in context: A068111 A162444 A052468 this_sequence A001790 A057908 
               A120828
%Y A055786 Adjacent sequences: A055783 A055784 A055785 this_sequence A055787 A055788 
               A055789
%K A055786 nonn,frac,nice,easy
%O A055786 0,3
%A A055786 N. J. A. Sloane (njas(AT)research.att.com), Jul 13 2000
%E A055786 Edited by Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 2009

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