Congruences for the Fishburn numbers

The Fishburn numbers, ξ(n)ξ(n), are defined by a formal power series expansion∑n=0∞ξ(n)qn=1+∑n=1∞∏j=1n(1−(1−q)j). For half of the primes p  , there is a non-empty set of numbers T(p)T(p) lying in [0,p−1][0,p−1] such that if j∈T(p)j∈T(p), then for all n≥0n≥0,ξ(pn+j)≡0(modp).