A002595 Denominators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x). (Formerly M4233 N1768)

1, 6, 40, 112, 1152, 2816, 13312, 10240, 557056, 1245184, 5505024, 12058624, 104857600, 226492416, 973078528, 2080374784, 23622320128, 30064771072, 635655159808, 446676598784, 11269994184704, 23639499997184, 6597069766656

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.

Konrad Knopp, Theory and application of infinite series, Blackie & Son Limited, London and Glasgow, 1954. See p. 215.

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).

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 31, equation 31:6:1 at page 290.

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.

Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1994. See p. 54.

Formula

a(n) = denominator((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 *)

Prog

(PARI) a(n) = denominator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))); \\ Stefano Spezia, Dec 31 2024

Crossrefs

A055786(n) / a(n) = A001147(n) / ( A000165(n) * (2*n+1))

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

Sequence in context: A110424 A114079 A211065 * A263956 A229638 A210291

Adjacent sequences: A002592 A002593 A002594 * A002596 A002597 A002598

Keyword

nonn,frac,nice,easy

Author

N. J. A. Sloane

Status

approved