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 A055786 Numerators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x). 7
 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 (list; graph; refs; listen; history; text; internal format)
 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) = numer((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)) MATHEMATICA Numerator/@Select[CoefficientList[Series[ArcSin[x], {x, 0, 60}], x], #!=0&]  (* Harvey P. Dale, Apr 29 2011 *) 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: A261659 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

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Last modified October 14 07:31 EDT 2019. Contains 327995 sequences. (Running on oeis4.)