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User:Andrey Mitin
Andrey Mitin.
B.S. Computer Science, University of Minnesota (2012-2015):
https://myaccount.umn.edu/lookup?SET_INSTITUTION=UMNTC&type=Internet+ID&CN=mitin001
VP Engineering, Modus (2015-present):
https://www.linkedin.com/in/andrey-mitin-13824a8a/
Reason for joining OEIS:
A104007 is missing critical information that pertains to that sequence.
The comment from Terry D. Grant from Sep 24 2016 suggests that the sequence might be related to zeta functions of odd positive integers because the even terms of the sequence are related to zeta functions of even positive integers, so the odd terms could be related to zeta functions of odd positive integers.
This is misleading because both even and odd terms of A104007 are related to zeta functions of even positive integers.
A104007 is generated from the expansion of x^2*(1-exp(-2*x))^(-2), which is a function involving the trigonometric functions cot and csc, both of which have the series coefficient related to Bernoulli numbers, which are only related to zeta functions of even positive integers and not of odd positive integers.
To verify, consider running the following Mathematica code:
Denominator[
Function[{n}, Piecewise[{{1/2 (-1 + n) Zeta[n], Mod[n, 2] == 0}, {Zeta[-1 + n], Mod[n, 2] == 1}}]] /@ Range[0, 20]]
This shows that A104007 can be generated from just the zeta functions of even positive integers.
One can use the connection of the expansion of x^2*(1-exp(-2*x))^(-2) to Bernoulli numbers to prove that the piecewise function above generates A104007 for all positive integers.