OFFSET
1,1
COMMENTS
Either polynomial may be 0. - Andrew Howroyd, Jul 10 2018
EXAMPLE
From Andrew Howroyd, Jul 10 2018: (Start)
Case n=1: Except for x^2 + x + 1 all polynomials with degree <= 2 are represented:
0 = 0*1,
1 = 1*1,
x = 1*x,
x + 1 = 1*(x + 1),
x^2 = x*x,
x^2 + 1 = (x + 1)*(x + 1),
x^2 + x = x*(x + 1).
Case n=3: There are 128 polynomials with degree <= 6. From this must be subtracted those polynomials whose factorizations into irreducible polynomials have degrees in the set {(6), (5+1), (4+1+1), (2+2+2), (5), (4+1), (4)}. 48 of these exclusions include an irreducible factor with degree >= 4. The other exclusion is (x^2 + x + 1)^3 which cannot be represented as the product of two polynomials of degree <= 3. Then a(3) = 128 - 48 - 1 = 79.
(End)
PROG
(PARI) \\ here b(n) is A001037.
b(n)={sumdiv(n, d, moebius(d)*2^(n/d))/n}
PartitionProduct(p, f)={my(r=1, k=0); for(i=1, length(p), if(i==length(p) || p[i]!=p[i+1], r*=f(p[i], i-k); k=i)); r}
ok(p, n, r)={poldegree(Pol(prod(i=1, #p, 1 + x^p[i] + O(x*x^n)))) >= r}
a(n)={my(u=vector(n, i, b(i)), s=2^(n+1)); for(r=1, n, forpart(p=n+r, if(ok(p, n, r), s+=PartitionProduct(p, (t, e)->binomial(u[t]+e-1, e))), [1, n])); s} \\ Andrew Howroyd, Jul 10 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 19 2003
EXTENSIONS
a(9)-a(25) from Andrew Howroyd, Jul 10 2018
STATUS
approved