Moments of infinite convolutions of symmetric Bernoulli distributions

We study the infinite convolution of symmetric Bernoulli distributions associated to a parameter r. We obtain an explicit formula for the moments as a function of Bernoulli numbers and conditioned partitions. Applying this formula we obtain the moments as a quotient of polynomials in the parameter r. The leading coefficient of the numerator is related to the asymptotic behavior of the moments and, unexpectedly, this coefficients are the absolute values of Euler numbers.