OFFSET
1,1
COMMENTS
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) = multinomial(a, b, c, d) * multinomial(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.
Equivalently, a(n) is the number of 4-tuples (r, s, t, u) of integers such that 0 <= r, s, t, u <= p-1 and either H(r) = H(s) = H(t) = H(u) = H(r + s + t + u) mod p, where H(n) is the n-th harmonic number and p is the n-th prime, or r + s + t + u >= 2*p. Note that the former case here implies that r + s + t + u <= p-1, as otherwise the expression H(r + s + t + u) mod p would be undefined. This restriction shows why these two cases can never overlap.
EXAMPLE
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).
MATHEMATICA
Table[Length[
Table[If[Plus @@ k >= p, Nothing,
If[Equal @@
Expand[{HarmonicNumber[k[[1]]], HarmonicNumber[k[[2]]],
HarmonicNumber[k[[3]]], HarmonicNumber[k[[4]]],
HarmonicNumber[k[[1]] + k[[2]] + k[[3]] + k[[4]]]},
Modulus -> p], k, Nothing]], {k, Tuples[Range[p] - 1, 4]}]] +
p*(p - 1)*(p^2 - p - 1)/2, {p, Prime[Range[5]]}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Joshua Crisafi, Nov 20 2021
STATUS
approved