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

The corrected triple sum expression of Ali\v{s}auskas (1987) 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 Ali\v{s}auskas also related to result of Horme\ss and Junker 1999) is rearranged into three new different double sum expressions (related to the hypergeometric Kamp\'e de F\'eriet 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 Kamp\'e de F\'eriet $F_{1:3}^{1:4}$ 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).