Notes on the formula for odd n
Richard Turk, Plymouth MA
Email to N. J. A. Sloane, Nov 21 2019

Regarding my submission to OEIS A007389, the proposed formula was proved using Mathematica.  The odd values of A007389 are found by solving the equation x^9-x^7-1=0  and summing each of the 9 roots to an odd power.  The result is an integer that agrees with my proposed formula for odd values of A007389 with n>=9.  The formula is general and used for odd g values in the present case g = 9. In Mathematica the formula is expressed as;

Sum[(2*n/(n - 2*(g - 2)*j - (g - 2)))*Hypergeometric2F1[-(n - 2*g*j - g)/2, -(2 j + 1), 1, 1], {j, 0, Floor[(n - g)/(2*g)]}] which has been expressed in proper text notation in the entry. 

I have shown that for odd values of n beyond 60 the proposed formula agrees with the sum of roots of x^9-x^7-1=0  to the odd power n.  The formula is applicable to any odd value of n greater than 9 based on my calculations and could be used to extent the Table in A007389 to higher odd values.