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 A001896 Numerators of cosecant numbers {-2*(2^(2*n-1)-1)*Bernoulli(2*n)}; also of Bernoulli(2n,1/2) and Bernoulli(2n,1/4). (Formerly M4403 N1858) 22
 1, -1, 7, -31, 127, -2555, 1414477, -57337, 118518239, -5749691557, 91546277357, -1792042792463, 1982765468311237, -286994504449393, 3187598676787461083, -4625594554880206790555, 16555640865486520478399, -22142170099387402072897 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Cosecant number are given by the integral: (-Pi^2)^(-n)*int((log(x/(1-x)))^2*n,x=0..1) - Groux Roland, Nov 10 2009 |A001896(n)|*pi^(2n)/A001897(n) is the value of the multi zeta function z(2,2,...,2) with n 2's, where z(k_l,k_2,...,k_n) = sum_{i_n >= i_(n-1) >= ... >= i_1 >= 1}1/((i_1)^k_1 (i_2)^k_2 ... (i_n)^k_n). The proof is simple: start with the product expansion sin(pi x)/(pi x) = product_{r>=1}(1-x^2/r^2), take reciprocals, and expand the right side. The coefficient of x^(2n) is seen to be z(2,2,...,2) with n 2's. - David Callan, Aug 27 2014 REFERENCES H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 187. S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51. N. E. Nörlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 458. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008; Page 7, 3rd table, (B^sin)_1,n is identical to |A001896| / A001897. S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51. [Annotated scanned copy of pages 38-51 only, plus notes] D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649. N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463] FORMULA a(n) = numerator((-1)^(n+1)*(2*Pi)^(-2*n)*(2*n)!*Li_{2*n}(-1)). - Peter Luschny, Jun 29 2012 E.g.f. 2*x*exp(x)/(exp(2*x) - 1) = 1 - 1/3*x^2/2! + 7/15*x^4/4! - 31/21*x^6/6! + .... = sum {n >= 0} A001896(n)/A001897(n)*x^(2*n)/(2*n)!. - Peter Bala, Jul 18 2013 See A062715 for a method of obtaining the cosecant numbers from the square of Pascal's triangle. - Peter Bala, Jul 18 2013 EXAMPLE 1, -1/12, 7/240, -31/1344, 127/3840, -2555/33792, 1414477/5591040, -57337/49152, 118518239/16711680, ... = A001896/A033469 Cosecant numbers {-2*(2^(2*n-1)-1)*Bernoulli(2*n)} are 1, -1/3, 7/15, -31/21, 127/15, -2555/33, 1414477/1365, -57337/3, 118518239/255, -5749691557/399, 91546277357/165, -1792042792463/69, 1982765468311237/1365, -286994504449393/3, 3187598676787461083/435, ... = A001896/A001897. MAPLE [ seq(numer(bernoulli(2*n, 1/2)), n=0..20) ]; MATHEMATICA a[n_] := -2*(2^(2*n-1)-1)*BernoulliB[2*n]; Table[a[n], {n, 0, 20}] // Numerator (* Jean-François Alcover, Sep 11 2013 *) PROG (PARI) a(n) = numerator(-2*(2^(2*n-1)-1)*bernfrac(2*n)); \\ Michel Marcus, Mar 01 2015 (Sage) def A001896_list(len):     R, C = [1], [1]+[0]*(len-1)     for n in (1..len-1):         for k in range(n, 0, -1):             C[k] = C[k-1] / (8*k*(2*k+1))         C[0] = -sum(C[k] for k in (1..n))         R.append((C[0]*factorial(2*n)).numerator())     return R print A001896_list(18) # Peter Luschny, Feb 20 2016 CROSSREFS Cf. A001897, A033469, A036280, A132092...A132106, A062715, A145901. Sequence in context: A277002 A036282 A033474 * A262630 A180147 A044049 Adjacent sequences:  A001893 A001894 A001895 * A001897 A001898 A001899 KEYWORD sign,frac AUTHOR STATUS approved

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