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Search: id:A001896
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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(2n,1/2) and Bernoulli(2n,1/4).
(Formerly M4403 N1858)
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+0
5
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| 1, -1, 7, -31, 127, -2555, 1414477, -57337, 118518239, -5749691557, 91546277357, -1792042792463, 1982765468311237, -286994504449393, 3187598676787461083, -4625594554880206790555, 16555640865486520478399, -22142170099387402072897
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Cosecant number are given by the integral: (-Pi^2)^(-n)*int((ln(x/(1-x)))^2*n,x=0..1) [From Groux roland (roland.groux(AT)orange.fr), Nov 10 2009]
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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.
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
N. E. N\"{o}rlund, Vorlesungen \"{u}ber 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).
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LINKS
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Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, Page 7, 3rd table, (B^sin)_1,n is identical to |A001896| / A001897.
Index entries for sequences related to Bernoulli numbers.
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FORMULA
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with(numtheory); [ seq(bernoulli(2*n, 1/2), n=0..20) ];
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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, ... = 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.
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CROSSREFS
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Cf. A001897, A033469.
Cf. A132092-A132106.
Sequence in context: A083420 A036282 A033474 this_sequence A044049 A005825 A086901
Adjacent sequences: A001893 A001894 A001895 this_sequence A001897 A001898 A001899
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KEYWORD
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sign,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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