(Not an answer - but too large for a comment) here are recurrences for the first few n. I don't have time right now to look at them, but shouldn't be hard to spot a pattern...
(1) -> P n == reduce(+, [binomial(n, 2*k)*(1-x)^k for k in 0..n | 2*k Q n == reduce(+, [binomial(n, 2*k+1)*(1-x)^(k+1) for k in 0..n | 2*k+1 n:=3; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 10))
(3) [[f(n): - 4f(n + 1) + f(n) + 1= 0,f(0)= 1]]
(4) -> n:=4; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 10))
(4) [[f(n): - 8n f(n + 1) + 4n f(n) + n= 0,f(0)= 1]]
(5) -> n:=5; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 30))
1
(5) [[f(n): - 16f(n + 2) + 12f(n + 1) - f(n) + 1= 0,f(0)= 1,f(1)= -]]
2
(6) -> n:=6; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 30))
(6)
1
[[f(n): - 32n f(n + 2) + 32n f(n + 1) - 6n f(n) + n= 0,f(0)= 1,f(1)= -]]
2
(7) -> n:=7; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 30))
(7)
[
[f(n): 64f(n + 3) - 80f(n + 2) + 24f(n + 1) - f(n) - 1= 0, f(0)= 1,
1 3
f(1)= -, f(2)= -]
2 8
]
(8) -> n:=8; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 100))
(8)
[
[f(n): 128n f(n + 3) - 192n f(n + 2) + 80n f(n + 1) - 8n f(n) - n= 0,
1 3
f(0)= 1, f(1)= -, f(2)= -]
2 8
]
(9) -> n:=9; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 100))
(9)
[
[f(n): - 256f(n + 4) + 448f(n + 3) - 240f(n + 2) + 40f(n + 1) - f(n) + 1= 0
,
1 3 5
f(0)= 1, f(1)= -, f(2)= -, f(3)= --]
2 8 16
]
(10) -> n:=10; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 100))
(10)
[
[
f(n):
- 512n f(n + 4) + 1024n f(n + 3) - 672n f(n + 2) + 160n f(n + 1)
+
- 10n f(n) + n
=
0
,
1 3 5
f(0)= 1, f(1)= -, f(2)= -, f(3)= --]
2 8 16
]
(11) -> n:=11; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 100))
(11)
[
[
f(n):
1024f(n + 5) - 2304f(n + 4) + 1792f(n + 3) - 560f(n + 2) + 60f(n + 1)
+
- f(n) - 1
=
0
,
1 3 5
f(0)= 1, f(1)= -, f(2)= -, f(3)= --]
2 8 16
]
and, in case it helps, here is the (nonlinear) recurrence for the sequence of $P_n/Q_n$:
(19) -> guessRec([P n/Q n for n in 1..100])
1
(19) [[f(n): ((- x + 1)f(n) + 1)f(n + 1) - f(n) - 1= 0,f(0)= - -----]]
x - 1