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A001896
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Numerators of cosecant numbers -2*(2^(2*n - 1) - 1)*Bernoulli(2*n); also of Bernoulli(2*n, 1/2) and Bernoulli(2*n, 1/4).
(Formerly M4403 N1858)
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26
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1, -1, 7, -31, 127, -2555, 1414477, -57337, 118518239, -5749691557, 91546277357, -1792042792463, 1982765468311237, -286994504449393, 3187598676787461083, -4625594554880206790555, 16555640865486520478399, -22142170099387402072897
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OFFSET
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0,3
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COMMENTS
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|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
See A062715 for a method of obtaining the cosecant numbers from the square of Pascal's triangle. - Peter Bala, Jul 18 2013
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REFERENCES
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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.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.
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).
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LINKS
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Masanobu Kaneko, Maneka Pallewatta, and Hirofumi Tsumura, On Polycosecant Numbers, J. Integer Seq. 23 (2020), no. 6, 17 pp. See line k=1 of Table 1 p. 3.
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FORMULA
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a(n) = numerator((-Pi^2)^(-n)*Integral_{x=0..1} (log(x/(1-x)))^2*n). - Groux Roland, Nov 10 2009
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} a(n)/A001897(n)*x^(2*n)/(2*n)!. - Peter Bala, Jul 18 2013
a(n) = numerator((-1)^n*I(n)), where I(n) = 2*Pi*Integral_{z=-oo..oo} (z^n / (exp(-Pi*z) + exp(Pi*z)))^2. - Peter Luschny, Jul 25 2021
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EXAMPLE
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1, -1/12, 7/240, -31/1344, 127/3840, -2555/33792, 1414477/5591040, -57337/49152, 118518239/16711680, ... = a(n)/A033469(n).
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, ... = a(n)/A001897(n).
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MAPLE
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seq(numer(bernoulli(2*n, 1/2)), n=0..20);
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MATHEMATICA
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a[n_] := -2*(2^(2*n-1)-1)*BernoulliB[2*n]; Table[a[n], {n, 0, 20}] // Numerator (* Jean-François Alcover, Sep 11 2013 *)
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PROG
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(PARI) a(n) = numerator(-2*(2^(2*n-1)-1)*bernfrac(2*n)); \\ Michel Marcus, Mar 01 2015
(Sage)
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
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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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