A036280 -id:A036280 - cp's OEIS frontend

A001896 Numerators of cosecant numbers -2(2^(2n - 1) - 1)Bernoulli(2n); also of Bernoulli(2n, 1/2) and Bernoulli(2n, 1/4).

1, -1, 7, -31, 127, -2555, 1414477, -57337, 118518239, -5749691557, 91546277357, -1792042792463, 1982765468311237, -286994504449393, 3187598676787461083, -4625594554880206790555, 16555640865486520478399, -22142170099387402072897

Offset: 0

N. J. A. Sloane

|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

			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).

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).

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-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
Index entries for sequences related to Bernoulli numbers.

Cf. A001897 (denominators), A033469, A036280, A062715, A145901.

Cf. A132092-A132099.

seq(numer(bernoulli(2*n, 1/2)), n=0..20);

a[n_] := -2*(2^(2*n-1)-1)*BernoulliB[2*n]; Table[a[n], {n, 0, 20}] // Numerator (* Jean-François Alcover, Sep 11 2013 *)

a(n) = numerator(-2*(2^(2*n-1)-1)*bernfrac(2*n)); \\ Michel Marcus, Mar 01 2015

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
A001896_list(18) # Peter Luschny, Feb 20 2016

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

A036283 Write cosec x = 1/x + Sum e_n x^(2n-1)/(2n-1)!; sequence gives denominators of e_n.

Original entry on oeis.org

6, 60, 126, 120, 66, 16380, 6, 4080, 7182, 3300, 138, 32760, 6, 1740, 42966, 8160, 6, 34545420, 6, 270600, 37926, 1380, 282, 1113840, 66, 3180, 21546, 3480, 354, 1703601900, 6, 16320, 194166, 60, 4686, 5043631320, 6, 60, 9954, 9200400, 498, 142981020, 6

Offset: 1

N. J. A. Sloane

Denominator of [2^(2n-1) - 1] * Bernoulli(2n)/n.
Equals the denominators of the LS1[-2*m,n=1] matrix coefficients of A160487 for m = 1, 2, ... - Johannes W. Meijer, May 24 2009
The products of the first n terms of this sequence appear in the denominators of the a(n) formulas of the right hand columns of triangle A161739. See A000292 (n=1), A107963 (n=2), A161740 (n=3) and A161741 (n=4). The next six values of n show that this pattern persists. - Johannes W. Meijer, Oct 22 2009

			x^(-1)+1/6*x+7/360*x^3+31/15120*x^5+...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).

Robert Israel, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).

Cf. A036280-A036282.

Cf. A000292, A006953, A107963, A160487, A161739, A161740, A161741.

seq(denom((2^(2*n-1)-1)*bernoulli(2*n)/n),n=1..100); # Robert Israel, Oct 14 2016

a(n) = denominator((2^(2*n-1)-1)*bernfrac(2*n)/n) \\ Hugo Pfoertner, Dec 18 2022

Apparently a(n) = 6*A202318(n). - Hugo Pfoertner, Dec 18 2022

Title corrected and offset changed by Johannes W. Meijer, May 21 2009

More terms, and edited by Robert Israel, Oct 14 2016

A036282 Write cosec x = 1/x + Sum_{n>=1} e_n * x^(2n-1)/(2n-1)!; sequence gives numerators of e_n.

Original entry on oeis.org

1, 7, 31, 127, 511, 1414477, 8191, 118518239, 5749691557, 91546277357, 162912981133, 1982765468311237, 22076500342261, 455371239541065869, 925118910976041358111, 16555640865486520478399, 1302480594081611886641, 904185845619475242495834469891

Offset: 1

N. J. A. Sloane

From Johannes W. Meijer, May 24 2009: (Start)
Absolute value of numerator of [2^(2n-1) - 1] * Bernoulli(2n)/n.
Equals the absolute values of the numerators of the LS1[ -2*m,n=1] matrix coefficients of A160487 for m = 1, 2, .. ,.
(End)

			cosec x
= x^(-1) + 1/6*x + 7/360*x^3 + 31/15120*x^5 + ...
= x^(-1) + 1/6 * x/1! + 7/60 * x^3/3! + 31/126 * x^5/5! + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).

Seiichi Manyama, Table of n, a(n) for n = 1..275
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).
R. P. Brent, Asymptotic approximation of central binomial coefficients with rigorous error bounds, arXiv:1608.04834 [math.NA], 2016.
Duane W. DeTemple, Shun-Hwa Wang, Half-integer approximations for the partial sums of harmonic series, J. Math. Anal. Applic. 160 (1991) 149-156
Simon Plouffe, On the values of the functions zeta and gamma, arXiv:1310.7195 [math.NT], 2013.
Eric Weisstein's World of Mathematics, Cosecant
Eric Weisstein's World of Mathematics, Riemann-Siegel Function
Wikipedia, Trigonometric functions

Cf. A036280, A036281, A036283.
Cf. A160487.
Differs from A282898.

a:= n-> (m-> numer(coeff(series(csc(x), x, m+1), x, m)*m!))(2*n-1):
seq(a(n), n=1..20);  # Alois P. Heinz, Jun 21 2018

a[n_] := Abs[ Numerator[ (2^(2*n-1)-1) * BernoulliB[2*n]/n ] ]; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, May 31 2013, after Johannes W. Meijer *)

a(n) = abs(numerator((2^(2*n-1)-1)*bernfrac(2*n)/n)); \\ Michel Marcus, Mar 01 2015

Title corrected and offset changed by Johannes W. Meijer, May 21 2009

A036281 Denominators in Taylor series for x * cosec(x).

Original entry on oeis.org

1, 6, 360, 15120, 604800, 3421440, 653837184000, 37362124800, 762187345920000, 2554547108585472000, 401428831349145600000, 143888775912161280000, 846912068365871834726400000, 93067260259985915904000000, 2706661834818276108533760000000

Offset: 0

N. J. A. Sloane

			cosec(x) = x^(-1)+1/6*x+7/360*x^3+31/15120*x^5+...
1, 1/6, 7/360, 31/15120, 127/604800, 73/3421440, 1414477/653837184000, 8191/37362124800, ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).
G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

Seiichi Manyama, Table of n, a(n) for n = 0..224 (terms 0..100 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 30.
J. Malenfant, Factorization of and Determinant Expressions for the Hypersums of Powers of Integers, arXiv preprint arXiv:1104.4332 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Hyperbolic Cosecant.
Eric Weisstein's World of Mathematics, Cosecant.
Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1994. See p. 54.
Index entries for Bernoulli numbers B(2n).

Cf. A036280, also A036282, A036283, B(2n) = A027641(2n) / A027642(2n).

Maple
```
series(csc(x),x,60);
```

a[n_] := 2(2^(2n-1)-1) Abs[BernoulliB[2n]]/(2n)! // Denominator;
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 14 2018 *)

def A036281_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] / (k*(4*k+2))
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0].denominator())
    return R
print(A036281_list(15)) # Peter Luschny, Feb 21 2016

A036280(n)/a(n)= 2 *(2^(2n-1) -1) *abs(B(2n)) / (2n)!.
From Arkadiusz Wesolowski, Oct 16 2013: (Start)
a(n) = A036280(n)*Pi^(2*n)/(zeta(2*n)*(2 - (2^(1-n))^2)).
a(n) = A230265(n)/2. (End)

A230265 Denominators of eta(2n)/Pi^(2n), where eta(n) is the Dirichlet eta function.

Original entry on oeis.org

2, 12, 720, 30240, 1209600, 6842880, 1307674368000, 74724249600, 1524374691840000, 5109094217170944000, 802857662698291200000, 287777551824322560000, 1693824136731743669452800000, 186134520519971831808000000

Offset: 0

Arkadiusz Wesolowski, Oct 14 2013

The first 5 terms of this sequence are the same as in A060055.

Wikipedia, Dirichlet eta function
Index entries for Bernoulli numbers B(2n)

Numerators give A036280.

for(n=0, 7, print1(2*denominator(polcoeff(Ser(1/sin(x)), 2*n-1)), ", "));

a(n) = A036280(n)*Pi^(2*n)/(zeta(2*n)*(1 - 2^(1-2*n))).
a(n) = denominator((-1)^(n+1)*BernoulliB(2*n)*(2^(2*n-1) - 1)/(2*n)!).
a(n) = 2*A036281(n).

A242032 A sequence related to lower bounds for the number of distinct differentiable structures on spheres of the form S^(4*k-1).

Original entry on oeis.org

2, 2, 7, 31, 127, 73, 1414477, 8191, 16931177, 5749691557, 91546277357, 3324754717, 1982765468311237, 22076500342261, 65053034220152267, 925118910976041358111, 16555640865486520478399, 8089941578146657681, 29167285342563717499865628061

Offset: 0

Peter Luschny, Aug 12 2014

nonn

Definition: Divide the prime factorization of an integer M into two parts: L(k, M) = product{p^v(M) the highest power of a prime p which divides M and p < k} and H(k, M) = M / L(k, M). Then a(n) = H(2*floor((n+1)/2), A246053(n)).
For some n the a(n) are lower bounds for the number of distinct differentiable structures on spheres. Compare theorem 2 of the Milnor 1959 paper which asserts that a(2) and a(4) through a(8) are lower bounds for the spheres S^(4*k-1) for k = 2 and 4,..,8.
Let b(n) = numerator(B(2*n)/(2*n)!*(4^n-2)*(-1)^(n-1)), B(n) Bernoulli number. Apart from n=0 and n=1 the first n such that a(n) != b(n) is n = 1437. Thus in the range [2..1436] the a(n) are the numerators in the Taylor series for x*cosec(x), A036280.

			a( 0) = 2
a( 1) = 2
a( 2) = 7
a( 3) = 31
a( 4) = 127
a( 5) = 73
a( 6) = 23 * 89 * 691
a( 7) = 8191
a( 8) = 31 * 151 * 3617
a( 9) = 43867 * 131071
a(10) = 283 * 617 * 524287
a(11) = 127 * 131 * 337 * 593
a(12) = 47 * 103 * 178481 * 2294797
a(13) = 31 * 601 * 1801 * 657931

Helaman Rolfe Pratt Ferguson, Bernoulli Numbers and Non-Standard Differentiable Structures on (4k-1)-Spheres, The Fibonacci Quarterly, Vol. 11(1), 1973.
Friedrich Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, 1956, Springer Berlin.
Friedrich Hirzebruch, Singularities and exotic spheres Séminaire Bourbaki, 10 (1966-1968), Exp. No. 314.
John Milnor, Differentiable Structures on Spheres, American Journal of Mathematics, Vol. 81, No. 4 (Oct., 1959), pp. 962-972.
John W. Milnor and Michel A. Kervaire, Bernoulli numbers, homotopy groups, and a theorem of Rohlin, J. A. Todd (ed.) , Proc. Internat. Congress Mathematicians (Edinburgh, 1958) , Cambridge Univ. Press (1960) pp. 454-458.
Dinesh S. Thakur, A note on numerators of Bernoulli numbers, Proc. Amer. Math. Soc. 140 (2012), 3673-3676.
Wikipedia, Exotic sphere

Cf. A036280, A246053, A240978.

h[x_] := Zeta[2x] (4^x-2);
a[n_] := Module[{M, k, p}, M = Denominator[h[Quotient[n+1, 2]] h[Quotient[ n, 2]]/h[n]]; k = 2 Quotient[n+1, 2]; p = 2; While[p < k, While[ Divisible[M, p], M = M/p]; p = NextPrime[p]]; M];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 12 2019, from Sage code *)

def A242032(n):
    h = lambda x: zeta(2*x)*(4^x-2)
    M = Integer((h((n+1)//2)*h(n//2)/h(n)).denominator())
    k = 2*((n+1)//2)
    P = Primes()
    p = P.first()
    while p < k:
        while p.divides(M):
            M /= p
        p = P.next(p)
    return M
[A242032(n) for n in (0..30)]

cp's OEIS Frontend

A001896 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).

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A036283 Write cosec x = 1/x + Sum e_n x^(2n-1)/(2n-1)!; sequence gives denominators of e_n.

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A036282 Write cosec x = 1/x + Sum_{n>=1} e_n * x^(2n-1)/(2n-1)!; sequence gives numerators of e_n.

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A036281 Denominators in Taylor series for x * cosec(x).

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Mathematica

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A230265 Denominators of eta(2*n)/Pi^(2*n), where eta(n) is the Dirichlet eta function.

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A242032 A sequence related to lower bounds for the number of distinct differentiable structures on spheres of the form S^(4*k-1).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

A001896 Numerators of cosecant numbers -2(2^(2n - 1) - 1)Bernoulli(2n); also of Bernoulli(2n, 1/2) and Bernoulli(2n, 1/4).

A230265 Denominators of eta(2n)/Pi^(2n), where eta(n) is the Dirichlet eta function.