|
%I #24 Aug 17 2020 23:04:00
%S 4,2,12,6,60,90,378,945,2700,9450,20790,93555,116093250,638512875,
%T 1403325,18243225,43418875500,325641566250,4585799468250,
%U 38979295480125,161192575293750,1531329465290625,640374140030625,13447856940643125,17558223649022306250
%N Denominators of coefficients in expansion of x^2*(1-exp(-2*x))^(-2).
%C 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.
%C Hyperbolic Trigonometric Functions:
%C Entire Series: x*(csch(x)^4) / 4*(coth(x)-1)^2).
%C Even Powers: (1/2)*(1-x*coth(x)).
%C Odd Powers: (1/4)*(2x + (csch(x)^2) + 2).
%C Circular Trigonometric Functions:
%C Entire Series: x*((csc(x)^2)/4) - cot(x)/2) + 1.
%C Even Powers: (1/2)*(1-x*cot(x)).
%C Odd Powers: (1/4)*(2x + (csc(x)^2) + 2).
%C 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
%C 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
%t Denominator[ CoefficientList[ Series[x^2*(1 - E^(-2x))^(-2), {x, 0, 33}], x]] (* _Robert G. Wilson v_, Apr 20 2005 *)
%t Denominator[
%t Function[{n},
%t Piecewise[{{1/2 (-1 + n) Zeta[n], Mod[n, 2] == 0}, {Zeta[-1 + n],
%t Mod[n, 2] == 1}}]] /@ Range[0, 20]] (* _Andrey Mitin_, Aug 16 2020 *)
%Y See A098087 for further information.
%Y Cf. A002432.
%K nonn,frac
%O 0,1
%A _N. J. A. Sloane_, Apr 17 2005
|
