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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

Integer component of the numerator of a close variant of Euler's infinite prime product zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)-1), namely with all minus signs changed into plus signs, as follows: zeta(4n)/zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)+1). The transcendental component is Pi^(2n).

For a detailed account of the results, including proof and relation to the zeta function, see Links for the PDF file submitted as supporting material.

The reference to Apostol is to a discussion of the equivalence of 1) zeta(2s)/zeta(s) and 2) a related infinite prime product, that is, Product_{sigma>1} prime(n)^s/(prime(n)^s + 1), with s being a complex variable such that s = sigma + i*t where sigma and t are real (following Riemann), using a type of proof different from the one posted below involving zeta(4n)/zeta(2n). On this, see also Hardy and Wright cited below. - Leo Depuydt, Nov 22 2013, Nov 27 2013

The background of the sequence is now described in the link below to L. Depuydt, The Prime Sequence ... . - Leo Depuydt, Aug 22 2014

From Robert Israel, Aug 22 2014: (Start)

Numerator of (-1)^n*B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!), where B(n) are the Bernoulli numbers (see A027641 and A027642).

Not the same as abs(A001067(2*n)): they differ first at n=17.

(End)

Denominator of a close variant of Euler's infinite prime product zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)-1), namely with all minus signs changed into plus signs, as follows: zeta(4n)/zeta(2n) = Product_{k>=1} prime(k)^(2n)/(prime(k)^(2n)+1).

For a detailed account of the results in question, including proof and relation to the zeta function, see the PDF file submitted as supporting material in A231273.

The reference to Apostol below is a discussion of the equivalence of 1) zeta(2s)/zeta(s) and 2) a related infinite prime product, that is, Product_{sigma>1} prime(n)^s/(prime(n)^s + 1), with s being a complex variable such that s = sigma + i*t where sigma and t are real (following Riemann), using a type of proof different from the one posted below involving zeta(4n)/zeta(2n). - Leo Depuydt, Nov 22 2013

Denominator of B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!) where B(n) are the Bernoulli numbers (see A027641 and A027642). - Robert Israel, Aug 22 2014