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

0,3

Comments

Note that the sequence is not monotonic.

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

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

Example

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))

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 *)

Prog

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

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

Sequence in context: A346715 A259853 A052468 * A001790 A173092 A057908

Adjacent sequences: A055783 A055784 A055785 * A055787 A055788 A055789

Keyword

nonn,frac,nice,easy

Author

N. J. A. Sloane, Jul 13 2000

Extensions

Edited by Johannes W. Meijer, Jul 06 2009

Status

approved