(convoluted) convolved Fibonacci numbers

[Pieter Moree]

Dear list,

(The material below is related to e.g. sequences
A00096, A001628, A001870 and A006504)

The convolved Fibonacci numbers can be defined by the generating
series

(1-z-z^2)^{-r}=\sum_{j>=0}F_{j+1}^{(r)}z^j.

Let r>=1 be a natural number. Define the convoluted convolved Fibonacci
numbers G_j^{(r)} by

{1\over r}sum_{d|n}mu(d)/(1-z^d-z^{2d})^{r/d}=\sum_{j>=0}G_{j+1}^{(r)}z^j.

Define the sign twisted convoluted convolved Fibonacci numbers
H_j^{(r)} by

{(-1)^r\over r}sum_{d|n}mu(d)(-1)^{r/d}/(1-z^d-z^{2d})^{r/d}=
\sum_{j>=0}H_{j+1}^{(r)}z^j.

(mu(n) denotes the Moebius function as usual).

The numbers H_j^{(r)} turn up in the evaluation of a certain constant
arising in number theory.

The numbers F_j^{(r)}, G_j^{(r)} and H_j^{(r)} are actually all
non-negative integers and have certain monotonicity properties in
both the j and r direction that are not apparent at all from the
definitions.

These three family of integer sequences are investigated in
ArXiv:math.CO/0311205, or alternatively

http://staff.science.uva.nl/~moree/fiboconv2.ps

I did not think about whether any of these (sub)sequences are relevant
for the OEIS and leave this to interested, if any, seqfans.

In this context I only made the remark (now in OEIS) that sequence
A006504 can be described by the values of a quartic polynomial.

Regards,
Pieter Moree
KdV-Institute
Amsterdam