A104007 Denominators of coefficients in expansion of x^2*(1-exp(-2*x))^(-2).

4, 2, 12, 6, 60, 90, 378, 945, 2700, 9450, 20790, 93555, 116093250, 638512875, 1403325, 18243225, 43418875500, 325641566250, 4585799468250, 38979295480125, 161192575293750, 1531329465290625, 640374140030625, 13447856940643125, 17558223649022306250

Offset: 0

Author

N. J. A. Sloane, Apr 17 2005

Keywords

• nonn
• frac

Comments

It appears that a(2n+2) = A002432(n). As A098087(n)/A104007(n) = x*(csch(x)^4)/(4*(coth(x)-1)^2), then a(2n+2) would represent the sequence of denominators for just the even powers of the full series representation at x=0. A002432 could be conjectured to be the non-hyperbolic, or circle trigonometric, function equivalent where the full series of denominators could be found by the formula x*((csc(x)^2)/4) - cot(x)/2) + 1 for a(n) > 4.

Hyperbolic Trigonometric Functions:

Entire Series: x*(csch(x)^4) / (4*(coth(x)-1)^2).

Even Powers: (1/2)*(1-x*coth(x)).

Odd Powers: (1/4)*(2x + (csch(x)^2) + 2).

Circular Trigonometric Functions:

Entire Series: x*((csc(x)^2)/4) - cot(x)/2) + 1.

Even Powers: (1/2)*(1-x*cot(x)).

Odd Powers: (1/4)*(2x + (csc(x)^2) + 2).

In turn, one may be able to derive some constant for x that can represent the zeta functions of odd positive integers. For zeta functions of even positive integers, that constant is Pi. - Terry D. Grant, Sep 24 2016

One can use the connection of the expansion of x^2*(1-exp(-2*x))^(-2) to Bernoulli numbers to prove that a(2n+2) = A002432(n), a(2n) = denominator(zeta(2n-2)) and a(2n-1) = denominator(1/2 (2n-3) zeta(2n-2)), and more generally that the expansion of x^2*(1-exp(-2*x))^(-2) is related to zeta(2n). The connection to Bernoulli numbers comes from the fact that x^2*(1-exp(-2*x))^(-2) is related to the trigonometric functions cot and csc, and they both have the series coefficients related to Bernoulli numbers, which are only related to zeta(2n), zeta functions of even positive integers, and not zeta(2n-1), zeta functions of odd positive integers. Because both a(2n) and a(2n-1) are related to zeta functions of even positive integers, the odd or even terms of this sequence are only related to zeta functions of odd positive integers if zeta(2n) is itself related to zeta(2n-1). - Andrey Mitin, Aug 16 2020

Crossrefs

See A098087 for further information.

Cf. A002432.

Programs

Mathematica

Denominator[ CoefficientList[ Series[x^2*(1 - E^(-2x))^(-2), {x, 0, 33}], x]] (* Robert G. Wilson v, Apr 20 2005 *)
Denominator[
 Function[{n},
   Piecewise[{{1/2 (-1 + n) Zeta[n], Mod[n, 2] == 0}, {Zeta[-1 + n],
Mod[n, 2] == 1}}]] /@ Range[0, 20]] (* Andrey Mitin, Aug 16 2020 *)