Writing t/(2n)-arctan(2*t/(4n+1)) as a series in t and then summing each coefficient as a series in n (in Maple) gives
(2-3/2*ln(2)-1/4*Pi)*t-1/48*Psi(2,5/4)*t^3+1/3840*Psi(4,5/4)*t^5-1/645120*Psi(6,5/4)*t^7+1/185794560*Psi(8,5/4)*t^9+O(t^11)
Psi(n,x) is the nth derivative of Psi(x), so it seems...