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 A104007 Denominators of coefficients in expansion of x^2*(1-exp(-2*x))^(-2). 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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 LINKS MATHEMATICA Denominator[ CoefficientList[ Series[x^2*(1 - E^(-2x))^(-2), {x, 0, 33}], x]] (* Robert G. Wilson v, Apr 20 2005 *) CROSSREFS See A098087 for further information. Cf. A002432. Sequence in context: A260434 A243344 A201825 * A191441 A152664 A167591 Adjacent sequences:  A104004 A104005 A104006 * A104008 A104009 A104010 KEYWORD nonn,frac AUTHOR N. J. A. Sloane, Apr 17 2005 STATUS approved

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