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RP Stanley in [S] observed that the series f (x)= C,“=,(c)’xn is transcendental over C (x) for even t> 1 and stated that it is unknown for odd t> 1 whether or not it is transcendental. We show here that (the reduction of) f is algebraic over any field of positive characteristic p and we then deduce (from the explicit equations obtained) that f is transcendental over any field of characteristic zero for any integer r> 1. We also give a generalisation of this result in the case of multinomial coefficients.

Note that if t= 1 then f is algebraic of degree at most 2 over any field (in fact f=(1-4x)-‘/*) and that for all t 2 1, f= 1 over any field of characteristic 2 so that we may suppose that, in the case of positive characteristic PI P> 2.