A036282 Write cosec x = 1/x + Sum_{n>=1} e_n * x^(2n-1)/(2n-1)!; sequence gives numerators of e_n.
1, 7, 31, 127, 511, 1414477, 8191, 118518239, 5749691557, 91546277357, 162912981133, 1982765468311237, 22076500342261, 455371239541065869, 925118910976041358111, 16555640865486520478399, 1302480594081611886641, 904185845619475242495834469891
Offset: 1
Examples
cosec x = x^(-1) + 1/6*x + 7/360*x^3 + 31/15120*x^5 + ... = x^(-1) + 1/6 * x/1! + 7/60 * x^3/3! + 31/126 * x^5/5! + ...
References
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..275
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).
- R. P. Brent, Asymptotic approximation of central binomial coefficients with rigorous error bounds, arXiv:1608.04834 [math.NA], 2016.
- Duane W. DeTemple, Shun-Hwa Wang, Half-integer approximations for the partial sums of harmonic series, J. Math. Anal. Applic. 160 (1991) 149-156
- Simon Plouffe, On the values of the functions zeta and gamma, arXiv:1310.7195 [math.NT], 2013.
- Eric Weisstein's World of Mathematics, Cosecant
- Eric Weisstein's World of Mathematics, Riemann-Siegel Function
- Wikipedia, Trigonometric functions
Programs
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Maple
a:= n-> (m-> numer(coeff(series(csc(x), x, m+1), x, m)*m!))(2*n-1): seq(a(n), n=1..20); # Alois P. Heinz, Jun 21 2018
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Mathematica
a[n_] := Abs[ Numerator[ (2^(2*n-1)-1) * BernoulliB[2*n]/n ] ]; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, May 31 2013, after Johannes W. Meijer *) -
PARI
a(n) = abs(numerator((2^(2*n-1)-1)*bernfrac(2*n)/n)); \\ Michel Marcus, Mar 01 2015
Extensions
Title corrected and offset changed by Johannes W. Meijer, May 21 2009
Comments