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A385458 Triangle read by rows: T(n,k) = exponent of the highest power of 2 dividing each Fibonomial coefficient fibonomial(n, k).

%I A385458 #39 Jul 18 2025 06:22:52
%S A385458 0,0,0,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0,3,3,2,3,3,0,0,0,3,2,2,3,
%T A385458 0,0,0,0,0,2,2,2,0,0,0,0,1,1,0,3,3,0,1,1,0,0,0,1,0,0,3,0,0,1,0,0,0,0,
%U A385458 0,0,0,0,0,0,0,0,0,0,0,4,4,3,4,4,1,4,4,3,4,4,0
%N A385458 Triangle read by rows: T(n,k) = exponent of the highest power of 2 dividing each Fibonomial coefficient fibonomial(n, k).
%H A385458 Paolo Xausa, <a href="/A385458/b385458.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).
%H A385458 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 A385458 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.
%H A385458 David Radcliffe, <a href="/A385458/a385458.png">Matrix plot of the first 768 rows</a>
%H A385458 Diana L. Wells, <a href="https://www.fq.math.ca/Scanned/32-2/wells.pdf">The Fibonacci and Lucas triangles modulo 2</a>, Fibonacci Quart. 32, no. 2 (1994), 111-123. (Theorem 2)
%F A385458 T(n, k) = A007814(A010048(n, k)).
%F A385458 T(n, k) = Sum_{i=1..k} (A337923(n+1-i) - A337923(i)).
%F A385458 T(n, k) = b(n) - b(k) - b(n - k), where b(n) = 2*floor(n/3) + floor(n/6) - A000120(floor(n/3)) = A385608(n) is the 2-adic valuation of the product of the first n Fibonacci numbers.
%F A385458 sign(T(n, k)) = 1 - A385456(n, k). - _Peter Luschny_, Jul 03 2025
%e A385458 Triangle begins:
%e A385458    n\k  0  1  2  3  4  5  6  7  8  9 10 11 12
%e A385458    0:   0
%e A385458    1:   0  0
%e A385458    2:   0  0  0
%e A385458    3:   0  1  1  0
%e A385458    4:   0  0  1  0  0
%e A385458    5:   0  0  0  0  0  0
%e A385458    6:   0  3  3  2  3  3  0
%e A385458    7:   0  0  3  2  2  3  0  0
%e A385458    8:   0  0  0  2  2  2  0  0  0
%e A385458    9:   0  1  1  0  3  3  0  1  1  0
%e A385458   10:   0  0  1  0  0  3  0  0  1  0  0
%e A385458   11:   0  0  0  0  0  0  0  0  0  0  0  0
%e A385458   12:   0  4  4  3  4  4  1  4  4  3  4  4  0
%t A385458 A385608[n_] := A385608[n] = 2*# + Quotient[n, 6] - DigitSum[#, 2] & [Quotient[n, 3]];
%t A385458 A385458[n_, k_] := A385608[n] - A385608[k] - A385608[n-k];
%t A385458 Table[A385458[n, k], {n, 0, 15}, {k, 0, n}] (* _Paolo Xausa_, Jul 04 2025 *)
%o A385458 (Python)
%o A385458 def b(n): return 2*(n//3) + n//6 - (n//3).bit_count()
%o A385458 def T(n, k): return b(n) - b(k) - b(n-k) # _David Radcliffe_, Jul 01 2025
%o A385458 (Julia)
%o A385458 function T_row(n)
%o A385458     function T(n, k)
%o A385458         c(a, b) = 2 * a + b ÷ 6 - count_ones(a)
%o A385458         (nd, nm) = divrem(n, 3)
%o A385458         (kd, km) = divrem(k, 3)
%o A385458         !(nm < km || (kd & (nd - kd)) != 0) && return 0
%o A385458         c(nd, n) - c(kd, k) - c((n - k) ÷ 3, n - k)
%o A385458     end
%o A385458     [T(n, k) for k in 0:n]
%o A385458 end
%o A385458 for n in 0:12 println(T_row(n)) end  # _Peter Luschny_, Jul 02 2025
%Y A385458 Cf. A000120, A010048, A007814, A065040, A337923, A385456, A385457, A385608.
%K A385458 nonn,tabl
%O A385458 0,23
%A A385458 _David Radcliffe_, Jun 29 2025