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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

Table of n, a(n) for n=0..24.

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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Last modified January 23 03:33 EST 2017. Contains 281185 sequences.