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Number of 5-tuples of nonnegative integers less than p for which 5-argument multinomial coefficients support a Lucas congruence modulo p^2, where p is the n-th prime.
2

%I #22 Dec 27 2021 11:05:02

%S 7,102,1759,10556,110328,259749,1019666,1793683,4721239,15226863,

%T 21315425,51979894,87138715,110728907,173179449,316739537,542780536,

%U 641673127,1027613847,1374672689,1580260645,2348585734,3008755650,4269518864,6573242285,8049042006

%N Number of 5-tuples of nonnegative integers less than p for which 5-argument multinomial coefficients support a Lucas congruence modulo p^2, where p is the n-th prime.

%C 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, p*e + v) = multinomial(a, b, c, d, e) * multinomial(r, s, t, u, v) mod p for all a >= 0, b >= 0, c >= 0, d >= 0, e >= 0, and r, s, t, u, v in the set {0, 1, ..., p-1}. a(n) is the number of such 5-tuples (r, s, t, u, v) for which this congruence also holds modulo p^2 for all a >= 0, b >= 0, c >= 0, d >= 0, and e >= 0, where p is the n-th prime.

%C Equivalently, a(n) is the number of 5-tuples (r, s, t, u, v) of integers such that 0 <= r, s, t, u, v <= p - 1 and either H(r) = H(s) = H(t) = H(u) = H(v) = H(r + s + t + u + v) mod p, where H(n) is the n-th harmonic number and p is the n-th prime, or r + s + t + u + v >= 2*p. Note that the former case here implies that r + s + t + u + v <= p-1, as otherwise the expression H(r + s + t + u + v) mod p would be undefined. This restriction shows why these two cases can never overlap.

%e For n = 1, p will be 2, and there are exactly a(1)=7 5-tuples of the form (r, s, t, u, v) that satisfy the desired properties that 0 <= r, s, t, u, v <= 1 and either H(r) = H(s) = H(t) = H(u) = H(v) = H(r + s + t + u + v) mod 2, where H(n) is the n-th harmonic number, or r + s + t + u + v >= 4: (0, 0, 0, 0, 0), (0, 1, 1, 1, 1), (1, 0, 1, 1, 1), (1, 1, 0, 1, 1), (1, 1, 1, 0, 1), (1, 1, 1, 1, 0), and (1, 1, 1, 1, 1).

%t Table[Length[Table[

%t If[Plus @@ k >= p, Nothing,

%t If[Equal @@

%t Expand[{HarmonicNumber[k[[1]]], HarmonicNumber[k[[2]]],

%t HarmonicNumber[k[[3]]], HarmonicNumber[k[[4]]],

%t HarmonicNumber[k[[5]]],

%t HarmonicNumber[k[[1]] + k[[2]] + k[[3]] + k[[4]] + k[[5]]]},

%t Modulus -> p], k, Nothing]], {k, Tuples[Range[p] - 1, 5]}]]

%t + p*(p - 1)*(93*p^3 - 17*p^2 - 122*p - 72)/120, {p, Prime[Range[5]]}]

%Y Cf. A348884, A348883, A348886, A348885, A001008, A002805.

%K nonn

%O 1,1

%A _Joshua Crisafi_, Nov 23 2021