OFFSET
0,2
COMMENTS
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.
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 + ...).
arccsc(x) = 1/x+1/(6*x^3)+3/(40*x^5)+5/(112*x^7)+35/(1152*x^9)+63/(2816*x^11)+...
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)+...)
arcsinh(x) = x-1/6*x^3+3/40*x^5-5/112*x^7+35/1152*x^9-63/2816*x^11+...
arccsc(x) = arcsin(1/x) and arcsec(x) = arccos(1/x): 1 < |x|
arccsch(x) = arcsinh(1/x) for 1 < |x|
Also denominator of (2n-1)!! / ((2n+1)*(2n)!!) (n=>0).
REFERENCES
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).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, Chap. 3.
Focus, vol. 16, no. 5, page 32, Oct 1996.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n=0..200
H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.
Eric Weisstein's World of Mathematics, Inverse Cosecant
Eric Weisstein's World of Mathematics, Inverse Cosine
Eric Weisstein's World of Mathematics, Inverse Secant
Eric Weisstein's World of Mathematics, Inverse Sine
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosecant
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Sine
Eric Weisstein's World of Mathematics, Archimedes' Spiral
FORMULA
a(n) = denom((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))). - Johannes W. Meijer, Jul 06 2009
MATHEMATICA
Denominator[Take[CoefficientList[Series[ArcSin[x], {x, 0, 50}], x], {2, -1, 2}]] (* Harvey P. Dale, Aug 06 2012 *)
CROSSREFS
Cf. A162443 where BG1[ -3,n] = (-1)*A002595(n-1)/A055786(n-1) for n =>1. - Johannes W. Meijer, Jul 06 2009
a(n) = 2*A143582(n+1) for n>=1. - Filip Zaludek, Oct 25 2016
KEYWORD
nonn,frac,nice,easy
AUTHOR
STATUS
approved