conjecture on Wave Sequences

[wouter meeussen]

the GF for the top line of the n-wave sequence, cfr http://home.wxs.nl/~lamoen/wiskunde/wave.htm
is:
Table[1 + Nest[-q/(q + #) &, x, 2n] /. x -> 0 // Simplify, {n, 5}]

(1 - q)^(-1),
(1 - q)/(1 - 3*q + q^2),
(1 - 3*q + q^2)/(1 - 6*q + 5*q^2 - q^3),
(1 - 6*q + 5*q^2 - q^3)/(1 - 10*q + 15*q^2 - 7*q^3 + q^4),
(1 - 10*q + 15*q^2 - 7*q^3 + q^4)/(1 - 15*q + 35*q^2 - 28*q^3 + 9*q^4 - q^5)}

giving the sequences :
{1, 1, 1, 1, 1, 1, 1, 1},
{1, 2, 5, 13, 34, 89, 233, 610}, A001519
{1, 3, 14, 70, 353, 1782, 8997, 45425}, A038213
{1, 4, 30, 246, 2037, 16886, 139997, 1160693}, A038225
{1, 5, 55, 671, 8272, 102091, 1260143, 15554592}A038261

true? known? proven? extendible to higher n?
W.