%I #28 Jan 11 2017 09:06:54
%S 1,2,1,3,0,1,4,2,0,1,5,0,3,0,1,6,3,2,0,0,1,7,0,0,0,0,0,1,8,4,0,2,0,0,
%T 0,1,9,0,7,0,0,0,3,0,1,10,5,6,0,2,3,0,0,0,1,11,0,0,0,0,0,0,0,0,0,1,12,
%U 6,4,3,0,2,0,0,0,0,0,1,13,0,0,0,0,0,0,0,0,0,0,0,1,14,7,0,0,0,0,2,0,0,0,0,0,0,1,15,0,5,0,3,0,0,0,0,0,0,0,0,0,1
%N Triangular table for division in ring GF(2)[X]: T(n,k) = n/k, or 0 if k is not a divisor of n, where the binary expansion of each number defines the corresponding (0,1)-polynomial.
%C This is GF(2)[X] analog of A126988, using "carryless division in base-2" instead of ordinary division.
%C The triangular table T(n,k), n=1.., k=1..n is read by rows: T(1,1), T(2,1), T(2,2), T(3,1), T(3,2), T(3,3), etc.
%H Antti Karttunen, <a href="/A280499/b280499.txt">Table of n, a(n) for n = 1..32896; the first 256 rows of triangle</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H <a href="/index/Ge#GF2X">Index entries for sequences operating on polynomials in ring GF(2)[X]</a>
%F T(n,k) = the unique d such that A048720(d,k) = n, provided that such d exists, otherwise zero.
%e The first 17 rows of the triangle:
%e 1
%e 2 1
%e 3 0 1
%e 4 2 0 1
%e 5 0 3 0 1
%e 6 3 2 0 0 1
%e 7 0 0 0 0 0 1
%e 8 4 0 2 0 0 0 1
%e 9 0 7 0 0 0 3 0 1
%e 10 5 6 0 2 3 0 0 0 1
%e 11 0 0 0 0 0 0 0 0 0 1
%e 12 6 4 3 0 2 0 0 0 0 0 1
%e 13 0 0 0 0 0 0 0 0 0 0 0 1
%e 14 7 0 0 0 0 2 0 0 0 0 0 0 1
%e 15 0 5 0 3 0 0 0 0 0 0 0 0 0 1
%e 16 8 0 4 0 0 0 2 0 0 0 0 0 0 0 1
%e 17 0 15 0 5 0 0 0 0 0 0 0 0 0 3 0 1
%e -----------------------------------
%e 7 ("111" in binary) encodes polynomial X^2 + X + 1, which is irreducible over GF(2) (7 is in A014580), so it is divisible only by itself and 1, and thus T(7,1) = 7, T(7,k) = 0 for k=2..6 and T(7,7) = 1.
%e 9 ("1001" in binary) encodes polynomial X^3 + 1, which is factored over GF(2) as (X+1)(X^2 + X + 1), and thus T(9,3) = 7 and T(9,7) = 3 because the polynomial X + 1 is encoded by 3 ("11" in binary).
%o (Scheme) (define (A280499 n) (A280500bi (A002024 n) (A002260 n))) ;; Code for A280500bi given in A280500.
%Y Lower triangular region of square array A280500.
%Y Transpose: A280494.
%Y Cf. A014580, A048720, A126988, A178908, A280500, A280493 (the row sums).
%K nonn,base,tabl,look
%O 1,2
%A _Antti Karttunen_, Jan 09 2017