A146305[n_, m_] := 
 2 (2 m + 3)! ((4 n + 2 m + 1)!/m!/(m + 2)!/n!/(3 n + 2 m + 3)!);
BrownE[r_, n_, m_] := 
 Module[{j, s, p}, 
 Which[r < 1, Return[0], r == 1, Return[A146305[n, m]], 
 r == 2, {s, j} = QuotientRemainder[n, 2];
 If[EvenQ[m], Return[0]];
 p = (m + 1)/2;
 If[p > 0 && s >= 0, 
 Return[2 (2 p)! ((4 s + 2 p + 2 j - 1)!/p!/(p - 1)!/
 s!/(3 s + 2 p + 2 j)!)], Return[0]], 
 r == 3 && Mod[n, 3] == 0 && Mod[m, 3] == 0, s = n/3; p = m/3;
 If[p >= 0 && s >= 0, 
 Return[(2 p + 1)! ((4 s + 2 p)!/p!/p!/s!/(3 s + 2 p + 1)!)], 
 Return[0]], r >= 3, 
 If[Mod[n - 1, r] == 0 && Mod[m + 3, r] == 0, s = (n - 1)/r; 
 p = (m + 3)/r - 1;
 If[p >= 0 && s >= 0, 
 Return[(2 p + 2)! ((4 s + 2 p + 1)!/p!/(p + 1)!/
 s!/(3 s + 2 p + 2)!)], Return[0]], Return[0]], True, 
 Return[0]]];
BrownG[n_, m_] := 
 Sum[EulerPhi[s] BrownE[s, n, m], {s, Divisors[m + 3]}]/(m + 3);
Table[BrownG[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten
(* Jean-François Alcover, Mar 29 2020,after R. J. Mathar *)