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Number of 4-tuples of nonnegative integers less than p for which 4 -argument multinomial coefficients support a Lucas congruence modulo p^2, where p is the n-th prime.

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Number of 4-tuples of base-nonnegative integers less than p digits for which 4 argument multinomial coefficients support a Lucas congruence modulo p^2, where p is the n-th prime.

For n = 2, p will be 3, and there are exactly a(2) = 20 4-tuples of the form (r, s, t, u) that satisfy the desired properties that 0 <= r, s, t, u <= 2 and either H(r) = H(s) = H(t) = H(u) = H(r + s + t + u) mod 3, where H(n) is the n-th harmonic number, or r + s + t + u >= 6: (0, 0, 0, 0), (0, 0, 0, 2), (0, 0, 2, 0), (0, 2, 0, 0), (0, 2, 2, 2), (1, 1, 2, 2), (1, 2, 1, 2), (1, 2, 2, 1), (1, 2, 2, 2), (2, 0, 0, 0), (2, 0, 2, 2), (2, 1, 1, 2), (2, 1, 2, 1), (2, 1, 2, 2), (2, 2, 0, 2), (2, 2, 1, 1), (2, 2, 1, 2), (2, 2, 2, 0), (2, 2, 2, 1), and (2, 2, 2, 2).

Number of 4-tuples of base-p digits for which 4 argument multinomial coefficients support a Lucas congruence modulo p^2, where p is the n-th prime.

For n = 2, p will be 3, and there are exactly a(2) = 20 triples 4-tuples of the form (r, s, t, u) that satisfy the desired properties that 0 <= r, s, t, u <= 2 and either H(r) = H(s) = H(t) = H(u) = H(r + s + t + u) mod 3, where H(n) is the n-th harmonic number, or r + s + t + u >= 6.

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This sequence stems from the property of the multinomial function that states that multinomial(p*a + r, p*b + s, p*c + t, p*d + u) = binomialmultinomial(a, b, c, d) * binomialmultinomial(r, s, t, u) mod p for all a >= 0, b >= 0, c >= 0, d >= 0, and r, s, t, u in the set {0, 1, ..., p-1}. a(n) is the number of such 4-tuples (r, s, t, u) for which this congruence also holds modulo p^2 for all a >= 0, b >= 0, c >= 0, and d >= 0, where p is the n-th prime.

For n = 2, p will be 3, and there are exactly a(2) = 20 triples of the form (r, s, t, u) that satisfy the desired properties that 0 <= r, s, t, u <= 4 2 and either H(r) = H(s) = H(t) = H(u) = H(r + s + t + u) mod 3, where H(n) is the n-th harmonic number, or r + s + t + u >= 6.

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