A055786 Numerators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).

1, 1, 3, 5, 35, 63, 231, 143, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 9694845, 100180065, 116680311, 2268783825, 1472719325, 34461632205, 67282234305, 17534158031, 514589420475, 8061900920775, 5267108601573

Offset: 0

Author

N. J. A. Sloane, Jul 13 2000

Keywords

• nonn
• frac
• nice
• easy

Comments

Note that the sequence is not monotonic.

Examples

			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.
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).
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).
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).
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).
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).
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.
a(4) = 35 = 3*5*7*9 / gcd( 3*5*7*9, (2*4*6*8) * (2*4+1))
		

References

• Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, ch. 4.2.6
• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
• H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, Chap. 3.

Links

• 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 Cosine.
• Eric Weisstein's World of Mathematics, Inverse Hyperbolic Sine.
• Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1994. See p. 54.

Crossrefs

Cf. A002595.

a(n) / A002595(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

Programs

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

Maple

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

Mathematica

Numerator/@Select[CoefficientList[Series[ArcSin[x],{x,0,60}],x], #!=0&]  (* Harvey P. Dale, Apr 29 2011 *)

PARI

vector(25, n, numerator(2*n*binomial(2*n,n)/(4^n*(2*n-1)^2)) ) \\ G. C. Greubel, Jan 25 2020

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

Formula

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 ).

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

Extensions

Edited by Johannes W. Meijer, Jul 06 2009