This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A104007 #28 May 04 2025 04:56:51
%S A104007 4,2,12,6,60,90,378,945,2700,9450,20790,93555,116093250,638512875,
%T A104007 1403325,18243225,43418875500,325641566250,4585799468250,
%U A104007 38979295480125,161192575293750,1531329465290625,640374140030625,13447856940643125,17558223649022306250
%N A104007 Denominators of coefficients in expansion of x^2*(1-exp(-2*x))^(-2).
%C A104007 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 A104007 Hyperbolic Trigonometric Functions:
%C A104007 Entire Series: x*(csch(x)^4) / (4*(coth(x)-1)^2).
%C A104007 Even Powers: (1/2)*(1-x*coth(x)).
%C A104007 Odd Powers: (1/4)*(2x + (csch(x)^2) + 2).
%C A104007 Circular Trigonometric Functions:
%C A104007 Entire Series: x*((csc(x)^2)/4) - cot(x)/2) + 1.
%C A104007 Even Powers: (1/2)*(1-x*cot(x)).
%C A104007 Odd Powers: (1/4)*(2x + (csc(x)^2) + 2).
%C A104007 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 A104007 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 A104007 Denominator[ CoefficientList[ Series[x^2*(1 - E^(-2x))^(-2), {x, 0, 33}], x]] (* _Robert G. Wilson v_, Apr 20 2005 *)
%t A104007 Denominator[
%t A104007 Function[{n},
%t A104007 Piecewise[{{1/2 (-1 + n) Zeta[n], Mod[n, 2] == 0}, {Zeta[-1 + n],
%t A104007 Mod[n, 2] == 1}}]] /@ Range[0, 20]] (* _Andrey Mitin_, Aug 16 2020 *)
%Y A104007 See A098087 for further information.
%Y A104007 Cf. A002432.
%K A104007 nonn,frac
%O A104007 0,1
%A A104007 _N. J. A. Sloane_, Apr 17 2005