A230265 -id:A230265 - 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

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)

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

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