Define recursively the rational fractions R_n by: R_0(x)=1; R_{n+1}(x) = (R_n(x)*x/(1-x^2))'. 2*a(n) is the maximal integer that can be factored out of the numerator of R'_n -- staying with polynomials with integer coefficients.
Empirical observations: the prime factorizations of the a(n) follow a pattern: the 2-adic valuation of a(n) is the 2-adic valuation of n; the 3-adic valuation of a(n) is (n mod 2); for p a prime >= 5, the p-adic valuation of a(n) is 0 (if p-1 does not divide n), 1 (if p-1 divides n but p does not) or 2 (if both p-1 and p divide n). So, a(n) = 1 when n is odd, and the prime factorizations of a(n) for the first few even n are:
\ p|
\ | 2 3 5 7 11 13 17 19 23 29 31 37 41 43
n \|
---+-------------------------------------------
2 | 1 1 . . . . . . . . . . . .
4 | 2 1 1 . . . . . . . . . . .
6 | 1 1 . 1 . . . . . . . . . .
8 | 3 1 1 . . . . . . . . . . .
10 | 1 1 . . 1 . . . . . . . . .
12 | 2 1 1 1 . 1 . . . . . . . .
14 | 1 1 . . . . . . . . . . . .
16 | 4 1 1 . . . 1 . . . . . . .
18 | 1 1 . 1 . . . 1 . . . . . .
20 | 2 1 2 . 1 . . . . . . . . .
22 | 1 1 . . . . . . 1 . . . . .
24 | 3 1 1 1 . 1 . . . . . . . .
26 | 1 1 . . . . . . . . . . . .
28 | 2 1 1 . . . . . . 1 . . . .
30 | 1 1 . 1 1 . . . . . 1 . . .
32 | 5 1 1 . . . 1 . . . . . . .
34 | 1 1 . . . . . . . . . . . .
36 | 2 1 1 1 . 1 . 1 . . . 1 . .
38 | 1 1 . . . . . . . . . . . .
40 | 3 1 2 . 1 . . . . . . . 1 .
42 | 1 1 . 2 . . . . . . . . . 1
