# [seqfan] Sequences of polynomials with nonnegative integer coefficients.

Antti Karttunen antti.karttunen at gmail.com
Mon Aug 3 14:16:05 CEST 2015

The prime-shift sequence (function) A003961: "If n = Product p(k)^e(k)
then a(n) = Product p(k+1)^e(k)" is very useful in many contexts, also
when defining sequences that encode in their prime factorization
polynomials with only nonnegative integer coefficients. In that
encoding, the exponent of the k-th prime A000040(k) in the prime
factorization of a(n) indicates the coefficient of term x^(k-1) in the
n-th polynomial of the sequence.

For example, see
http://oeis.org/A206296 for Fibonacci polynomials
and
http://oeis.org/A260443 for Stern polynomials.

As you can see, A003961(a(n)) corresponds to multiplying the
polynomial encoded by a(n) with its variable x, and multiplying a(i) *
a(j) corresponds to adding together the polynomials encoded by a(i)
and a(j).

These codes of polynomials can then be evaluated at point x=1, 2, 3,
... by applying the row 1, 2, 3, ... of
http://oeis.org/A104244
to them, for example, the sequence of Fibonacci polynomials evaluated
at x=2 yields Pell numbers: A048675(A206296(n)) = A000129(n).

(Do we have/want similar tables for evaluating such polynomials at
x=-1, -2, -3, etc. ? Or at x = 1, 1/2, 1/3, 1/4, ??? Separate tables
and its rows for numerators and denominators of the results for this
latter).

Similarly, Stern polynomials evaluated at x=3 yields: A090880(a(n)) =
A178590(n) which has a scatter-plot reminiscent of some real fractals:
http://oeis.org/A178590/graph

(And it's not surprise that http://oeis.org/A260443/graph looks quite
same with log-display).

Now, what other such famous polynomial recurrences there are? E.g.,
Lucas polynomials? (Seems not yet to be in OEIS).

But you might as well roll a few such recurrences of your own! Just
implement A003961 as a function in your favourite CAS, and start
experimenting with such recurrences. The convention seems to be that
they should be named after the integer sequence obtained when
evaluated at x=1 (in this encoding, by counting the number of divisors
with bigomega, A001222).

Best,

Antti