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

%I A385434 #24 Jun 29 2025 12:22:11
%S A385434 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,
%T A385434 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,
%U A385434 2,2,1,1,1,1,2,2,1,1
%N A385434 Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2, reduced mod 3.
%C A385434 Row sums give A385435.
%H A385434 Donald E. Knuth and Herbert S. Wilf, <a href="https://www.math.upenn.edu/~wilf/website/dm36.pdf">The power of a prime that divides a generalized binomial coefficient</a>, J. Reine Angew. Math., 396:212-219, 1989.
%H A385434 Romeo Meštrović, <a href="https://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878--2014)</a>, arXiv preprint arXiv:1409.3820 [math.NT], 2014. (See Equation (80) on p. 31.)
%F A385434 a(n) = A022166(n) mod 3.
%F A385434 T(2n, 2k) = T(2n+1, 2k) = T(2n, 2k+1) = binomial(n, k) mod 3; T(2n, 2k+1) = 0.
%e A385434 Triangle begins:
%e A385434   [ 0] [1]
%e A385434   [ 1] [1, 1]
%e A385434   [ 2] [1, 0, 1]
%e A385434   [ 3] [1, 1, 1, 1]
%e A385434   [ 4] [1, 0, 2, 0, 1]
%e A385434   [ 5] [1, 1, 2, 2, 1, 1]
%e A385434   [ 6] [1, 0, 0, 0, 0, 0, 1]
%e A385434   [ 7] [1, 1, 0, 0, 0, 0, 1, 1]
%e A385434   [ 8] [1, 0, 1, 0, 0, 0, 1, 0, 1]
%e A385434   [ 9] [1, 1, 1, 1, 0, 0, 1, 1, 1, 1]
%e A385434   [10] [1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1]
%e A385434   [11] [1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1]
%t A385434 Table[Mod[QBinomial[n, k, 2],3], {n, 0, 11}, {k, 0, n}] // Flatten  (* _James C. McMahon_, Jun 29 2025 *)
%o A385434 (SageMath)
%o A385434 def T(n, k): return mod(gaussian_binomial(n, k).subs(q=2), 3)
%o A385434 for n in range(12): print([T(n, k) for k in range(n+1)])  # _Peter Luschny_, Jun 29 2025
%Y A385434 Cf. A007318, A022166, A385435.
%K A385434 nonn,tabl
%O A385434 0,13
%A A385434 _David Radcliffe_, Jun 28 2025