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Search: id:A002595
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| A002595 |
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Denominators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).
(Formerly M4233 N1768)
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+0
6
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| 1, 6, 40, 112, 1152, 2816, 13312, 10240, 557056, 1245184, 5505024, 12058624, 104857600, 226492416, 973078528, 2080374784, 23622320128, 30064771072, 635655159808, 446676598784, 11269994184704, 23639499997184, 6597069766656
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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|
arcsch(x) = arsinh(1/x) for 1 < |x|
Also denominator of (2n-1)!! / ((2n+1)*(2n)!!) (n=>0).
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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.
H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
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
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FORMULA
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Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 2009: (Start)
a(n) =denom((2*n)!/(2^(2*n)*(n)!^2*(2*n+1)))
(End)
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CROSSREFS
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A055786(n) / a(n) = A001147(n) / ( A000165(n) * (2*n+1))
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 2009: (Start)
Cf. A162443 where BG1[ -3,n] = (-1)*A002595(n-1)/A055786(n-1) for n =>1.
(End)
Sequence in context: A045565 A110424 A114079 this_sequence A089207 A027777 A073773
Adjacent sequences: A002592 A002593 A002594 this_sequence A002596 A002597 A002598
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KEYWORD
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nonn,frac,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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