A348887 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.

7, 102, 1759, 10556, 110328, 259749, 1019666, 1793683, 4721239, 15226863, 21315425, 51979894, 87138715, 110728907, 173179449, 316739537, 542780536, 641673127, 1027613847, 1374672689, 1580260645, 2348585734, 3008755650, 4269518864, 6573242285, 8049042006

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, 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.

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.

Links

Table of n, a(n) for n=1..26.

Example

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).

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[[5]]],
       HarmonicNumber[k[[1]] + k[[2]] + k[[3]] + k[[4]] + k[[5]]]},
      Modulus -> p], k, Nothing]], {k, Tuples[Range[p] - 1, 5]}]]
+ p*(p - 1)*(93*p^3 - 17*p^2 - 122*p - 72)/120, {p, Prime[Range[5]]}]

Crossrefs

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

Sequence in context: A357334 A329239 A020477 * A358159 A203356 A346719

Adjacent sequences: A348884 A348885 A348886 * A348888 A348889 A348890

Keyword

nonn

Author

Joshua Crisafi, Nov 23 2021

Status

approved