A385434 Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2, reduced mod 3.

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1

Offset

0,13

Comments

Row sums give A385435.

Links

Table of n, a(n) for n=0..77.

Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Romeo Meštrović, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014. (See Equation (80) on p. 31.)

Formula

a(n) = A022166(n) mod 3.

T(2n, 2k) = T(2n+1, 2k) = T(2n, 2k+1) = binomial(n, k) mod 3; T(2n, 2k+1) = 0.

Example

Triangle begins:
  [ 0] [1]
  [ 1] [1, 1]
  [ 2] [1, 0, 1]
  [ 3] [1, 1, 1, 1]
  [ 4] [1, 0, 2, 0, 1]
  [ 5] [1, 1, 2, 2, 1, 1]
  [ 6] [1, 0, 0, 0, 0, 0, 1]
  [ 7] [1, 1, 0, 0, 0, 0, 1, 1]
  [ 8] [1, 0, 1, 0, 0, 0, 1, 0, 1]
  [ 9] [1, 1, 1, 1, 0, 0, 1, 1, 1, 1]
  [10] [1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1]
  [11] [1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1]

Mathematica

Table[Mod[QBinomial[n, k, 2], 3], {n, 0, 11}, {k, 0, n}] // Flatten  (* James C. McMahon, Jun 29 2025 *)

Prog

(SageMath)
def T(n, k): return mod(gaussian_binomial(n, k).subs(q=2), 3)
for n in range(12): print([T(n, k) for k in range(n+1)])  # Peter Luschny, Jun 29 2025

Crossrefs

Cf. A007318, A022166, A385435.

Sequence in context: A268479 A035196 A287475 * A158020 A051160 A051159

Adjacent sequences: A385431 A385432 A385433 * A385435 A385436 A385437

Keyword

nonn,tabl

Author

David Radcliffe, Jun 28 2025

Status

approved