A036281 -id:A036281 - cp's OEIS frontend

A036280 Numerators in Taylor series for x * cosec(x).

1, 1, 7, 31, 127, 73, 1414477, 8191, 16931177, 5749691557, 91546277357, 3324754717, 1982765468311237, 22076500342261, 65053034220152267, 925118910976041358111, 16555640865486520478399, 8089941578146657681, 29167285342563717499865628061

Offset: 0

N. J. A. Sloane

These are also the numerators of the coefficients appearing in the Maclaurin summation formula (which might be called the 'Maclaurin numbers') (see Gould & Squire, p. 45). - Peter Luschny, Feb 20 2016

			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, ...

G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

Seiichi Manyama, Table of n, a(n) for n = 0..275 (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, p. 75 (4.3.68).
H. W. Gould and W. Squire, Maclaurin's second formula and its generalization, Amer. Math. Monthly, 70 (1963), pp. 44-52.
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.

Cf. A036281, A036282, A036283.

Cf. A230265.

series(x*csc(x),x,60);
seq(numer((-1)^n*bernoulli(2*n,1/2)/(2*n)!), n=0..30); # Robert Israel, Mar 21 2016

nn = 34; t = Numerator[CoefficientList[Series[x*Csc[x], {x, 0, nn}], x]*Range[0, nn]!]; Take[t, {1, nn-1, 2}] (* T. D. Noe, Oct 28 2013 *)

a(n):=num(sum(sum((2^(1-j)*(-1)^(n+j-1)*binomial(k,j)*sum((j-2*i)^(2*n+j-2)*binomial(j,i)*(-1)^(i),i,0,floor(j/2)))/(2*n+j-2)!,j,1,k),k,1,2*n-2)); /* n>1. a(1)=1. */ /* Vladimir Kruchinin, Apr 12 2011 */

a(n):=(sum((sum(binomial(j,2*k-1)*(j-1)!*2^(1-j)*(-1)^(n+1+j)*stirling2(2*n+1,j),j,2*k-1,2*n+1))/(2*k-1),k,1,n+1))/(2*n)!;
/* Vladimir Kruchinin, Mar 21 2016 */

a(n)=numerator(sum(k=1,n,sum(j=0,k/2,binomial(3*n,n-k)*(-1)^(n+j)*(2*j-k)^(2*n+k)*2^(n+1-k)*(n+1)!/(j!*(k-j)!*(k+1))))/((3*n)!*2^n))+(n==0) \\ Tani Akinari, Feb 22 2025

my(x='x+O('x^40), v=apply(numerator, Vec(x/sin(x)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Feb 23 2025

def A036280_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].numerator())
    return R
print(A036280_list(19)) # Peter Luschny, Feb 20 2016

Numerator of Sum_{k=1..2*n-2} Sum_{j=1..k} 2^(1-j)*(-1)^(n+j-1) * binomial(k,j) * Sum_{i=0..floor(j/2)} (j-2*i)^(2*n+j-2) * binomial(j,i) * (-1)^i/(2*n+j-2)!, n > 1. - Vladimir Kruchinin, Apr 12 2011
E.g.f.: x/sin(x) = 1 + (x^2/(6-x^2))*T(0), where T(k) = 1 - x^2*(2*k+2)*(2*k+3)/( x^2*(2*k+2)*(2*k+3) + ((2*k+2)*(2*k+3) - x^2)*((2*k+4)*(2*k+5) - x^2)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013
a(n) = numerator((-1)^n*B(2*n,1/2)/(2*n)!) where B(n,x) denotes the Bernoulli polynomial. - Peter Luschny, Feb 20 2016
a(n) = numerator(Sum_{k=1..n+1}((Sum_{j=2*k-1..2*n+1}(binomial(j,2*k-1)*(j-1)!*2^(1-j)*(-1)^(n+1+j)*stirling2(2*n+1,j)))/(2*k-1))/(2*n)!). - Vladimir Kruchinin, Mar 21 2016
a(n) = numerator(eta(2*n)/Pi^(2*n)), where eta(n) is the Dirichlet eta function. See A230265 for denominator. - Mohammed Yaseen, Aug 02 2023
a(n) = numerator((Sum_{k=1..n} Sum_{j=0..floor(k/2)} binomial(3*n,n-k)*(-1)^(n+j)*(2*j-k)^(2*n+k)*2^(n+1-k)*(n+1)!/(j!*(k-j)!*(k+1)))/((3*n)!*2^n)) for n > 0. - Tani Akinari, Feb 22 2025

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

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).

cp's OEIS Frontend

A036280 Numerators in Taylor series for x * cosec(x).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Maxima

PARI

PARI

Sage

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A165641 A091137(n) / A001316(n) .

Views

Author

Keywords

Crossrefs

Extensions

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