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

1,2

Comments

From Johannes W. Meijer, May 24 2009: (Start)

Absolute value of numerator of [2^(2n-1) - 1] * Bernoulli(2n)/n.

Equals the absolute values of the numerators of the LS1[ -2*m,n=1] matrix coefficients of A160487 for m = 1, 2, .. ,.

(End)

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

Example

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! + ...

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

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 *)

Prog

(PARI) a(n) = abs(numerator((2^(2*n-1)-1)*bernfrac(2*n)/n)); \\ Michel Marcus, Mar 01 2015

Crossrefs

Cf. A036280, A036281, A036283.

Cf. A160487.

Differs from A282898.

Sequence in context: A083420 A277002 A282898 * A033474 A001896 A262630

Adjacent sequences: A036279 A036280 A036281 * A036283 A036284 A036285

Keyword

nonn,frac,easy,changed

Author

N. J. A. Sloane

Extensions

Title corrected and offset changed by Johannes W. Meijer, May 21 2009

Status

approved