A001897 Denominators of cosecant numbers: -2*(2^(2*n-1)-1)*Bernoulli(2*n).

1, 3, 15, 21, 15, 33, 1365, 3, 255, 399, 165, 69, 1365, 3, 435, 7161, 255, 3, 959595, 3, 6765, 903, 345, 141, 23205, 33, 795, 399, 435, 177, 28393365, 3, 255, 32361, 15, 2343, 70050435, 3, 15, 1659, 115005, 249, 1702155, 3, 30705, 136059, 705, 3, 2250885, 3, 16665, 2163

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac

Comments

Same as half the denominators of the even-indexed Bernoulli numbers B_{2*n} for n>0, by the von Staudt-Clausen theorem and Fermat's little theorem. - Bernd C. Kellner and Jonathan Sondow, Jan 02 2017 [This is implemented in the second Maple program. - Peter Luschny, Aug 21 2021]

Examples

			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.
		

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.
• 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).

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]
• 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.
• 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 1924, p. 27.
• N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]

Crossrefs

Cf. A001896 (numerators), A027642, A033469, A160014, A241171, A277087, A340556.

Cf. A132092-A132099,

Programs

Magma

[Denominator(2*(1-2^(2*n-1))*Bernoulli(2*n)): n in [0..55]]; // G. C. Greubel, Apr 06 2019

Maple

b := n -> bernoulli(n)*2^add(i,i=convert(n,base,2));
a := n -> denom(b(2*n)); # Peter Luschny, May 02 2009
# Alternative :
Clausen := proc(n) local i,S; map(i->i+1, numtheory[divisors](n));
S := select(isprime, %); if S <> {} then mul(i,i=S) else NULL fi end:
A001897_list := n -> [1,seq(Clausen(2*i)/2,i=1..n-1)];
A001897_list(52); # Peter Luschny, Oct 03 2011

Mathematica

a[n_] := Denominator[-2*(2^(2*n-1)-1)*BernoulliB[2*n]]; Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Sep 11 2013 *)

PARI

a(n) = denominator(-2*(2^(2*n-1)-1)*bernfrac(2*n)); \\ Michel Marcus, Apr 06 2019

Sage

def A001897(n):
    if n == 0:
        return 1
    M = (d + 1 for d in divisors(2 * n))
    return prod(s for s in M if is_prime(s)) / 2
[A001897(n) for n in range(55)]  # Peter Luschny, Feb 20 2016

Formula

a(0)=1, a(n)=(1/2)*A002445(n) for n>=1. - Joerg Arndt, May 07 2012

a(n) = denominator((2*n)!*Li_{2*n}(1)) for n > 0. - Peter Luschny, Jun 29 2012

a(0)=1, a(n) = (1/2)*A027642(2*n) = (3/2)*A277087(n) for n>=1. - Jonathan Sondow, Dec 14 2016

From Peter Luschny, Sep 06 2017: (Start)

a(n) = denominator(r(n)) where r(n) = Sum_{0..n} (-1)^(n-k)*A241171(n, k)/(2*k+1).

a(n) = denominator(bernoulli(2*n, 1/2))/4^n = A033469(n)/4^n. (End)

Apparently a(n) = denominator(Sum_{k=0..2*n-2} (-1)^k*E2(2*n-1, k+1)/binomial(4*n-1, k+1)), where E2(n, k) denotes the second-order Eulerian numbers A340556. - Peter Luschny, Feb 17 2021