Congruences of Multiple Sums Involving Sequences Invariant Under the Binomial Transform

We prove several congruences modulo a power of a prime, such as S-0 < k1 < center dot center dot center dot < kn < p (p - k(n)/3) (-1)(k)(n) / k(1) center dot center dot center dot k(n) equivalent to {-2(n+1) + 2 / 6(n+1) p Bp-n-1 (1/3) (mod p(2)), if n is odd, -2(n+1) + 4 / n6(n) Bp-n (1/3) (mod p), if n is even, where n is a positive integer and p is a prime such that p > max(n + 1, 3).