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

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

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• nice
• easy

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).
• 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.

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

Programs

Mathematica

Denominator[Take[CoefficientList[Series[ArcSin[x],{x,0,50}],x],{2,-1,2}]] (* Harvey P. Dale, Aug 06 2012 *)

PARI

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

Formula

a(n) = denominator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))). - Johannes W. Meijer, Jul 06 2009