A001897 Denominators of cosecant numbers: -2*(2^(2*n-1)-1)*Bernoulli(2*n). (Formerly M2983 N1205)

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

0,2

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]

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

Table of n, a(n) for n=0..51.

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]

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

Example

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

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 *)

Prog

(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
(PARI) a(n) = denominator(-2*(2^(2*n-1)-1)*bernfrac(2*n)); \\ Michel Marcus, Apr 06 2019
(Magma) [Denominator(2*(1-2^(2*n-1))*Bernoulli(2*n)): n in [0..55]]; // G. C. Greubel, Apr 06 2019

Crossrefs

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

Cf. A132092-A132099,

Sequence in context: A097571 A048087 A316751 * A074214 A036897 A129966

Adjacent sequences: A001894 A001895 A001896 * A001898 A001899 A001900

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved