6j-symbols for symmetric representations of SO(n) as the double series

The corrected triple sum expression of Alisauskas (1987 J. Phys. A: Math. Gen. 20 35) for the recoupling (Racah) coefficients (6j-symbols) of the symmetric (most degenerate) representations of the orthogonal groups SO(n) (previously derived from the fourfold sum expression of Alisauskas also related to the result of Hormess and Junker (1999 J. Phys. A: Math. Gen. 32 4249) is rearranged into three new different double sum expressions (related to the hypergeometric Kampe de Feriet type series) and a new triple sum expression with preferable summation condition. The Regge type symmetry of special 6j-symbols of the orthogonal groups SO(n) in terms of special Kampe de Feriet F1:41:3 series is revealed. The recoupling coefficients for antisymmetric representations of symplectic group Sp(2n) are derived using their relation with the recoupling coefficients of the formal orthogonal group SO(-2n)