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%I #29 Feb 17 2022 10:01:10
%S 1,1,3,5,35,63,231,143,6435,12155,46189,88179,676039,1300075,5014575,
%T 9694845,100180065,116680311,2268783825,1472719325,34461632205,
%U 67282234305,17534158031,514589420475,8061900920775,5267108601573
%N Numerators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).
%C Note that the sequence is not monotonic.
%D Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, ch. 4.2.6
%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.
%H T. D. Noe, <a href="/A055786/b055786.txt">Table of n, a(n) for n=0..200</a>
%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/InverseHyperbolicCosine.html">Inverse Hyperbolic Cosine</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InverseHyperbolicSine.html">Inverse Hyperbolic Sine</a>
%F 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 a(n) = numerator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))). - _Johannes W. Meijer_, Jul 06 2009
%e 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 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 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 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 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 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 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 a(4) = 35 = 3*5*7*9 / gcd( 3*5*7*9, (2*4*6*8) * (2*4+1))
%p seq( numer( (n+1)*binomial(2*n+2,n+1)/(2^(2*n+1)*(2*n+1)^2) ), n=0..25); # _G. C. Greubel_, Jan 25 2020
%t Numerator/@Select[CoefficientList[Series[ArcSin[x],{x,0,60}],x], #!=0&] (* _Harvey P. Dale_, Apr 29 2011 *)
%o (PARI) vector(25, n, numerator(2*n*binomial(2*n,n)/(4^n*(2*n-1)^2)) ) \\ _G. C. Greubel_, Jan 25 2020
%o (Magma) [Numerator( (n+1)*Binomial(2*n+2,n+1)/(2^(2*n+1)*(2*n+1)^2) ): n in [0..25]]; // _G. C. Greubel_, Jan 25 2020
%o (Sage) [numerator( (n+1)*binomial(2*n+2,n+1)/(2^(2*n+1)*(2*n+1)^2) ) for n in (0..25)] # _G. C. Greubel_, Jan 25 2020
%Y Cf. A002595.
%Y a(n) / A002595(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
%K nonn,frac,nice,easy
%O 0,3
%A _N. J. A. Sloane_, Jul 13 2000
%E Edited by _Johannes W. Meijer_, Jul 06 2009
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