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A347563 Binomial complement triangle, T(n,k) = LCM(1,...,n)/binomial(n,k) for 0 <= k <= n, a(0) = T(0,0) = 0, read by rows.

%I A347563 #33 Oct 09 2021 16:07:38
%S A347563 0,1,1,2,1,2,6,2,2,6,12,3,2,3,12,60,12,6,6,12,60,60,10,4,3,4,10,60,
%T A347563 420,60,20,12,12,20,60,420,840,105,30,15,12,15,30,105,840,2520,280,70,
%U A347563 30,20,20,30,70,280,2520
%N A347563 Binomial complement triangle, T(n,k) = LCM(1,...,n)/binomial(n,k) for 0 <= k <= n, a(0) = T(0,0) = 0, read by rows.
%C A347563 The one's complement of each carry value, in base prime p, defined in Lucas's Theorem. Also works using Erdős's method (see formula below).
%C A347563 At row n of the triangle, the values are symmetrical with the largest values occurring at T(n,0) = T(n,n) = LCM(1,...,n). The smallest value(s) occur at k = n/2 when n is even, and at k = floor(n/2) and k = floor(n/2)+1 when n is odd. T(n,k) = T(n,n-k).
%C A347563 Conjecture: For all n, T(n,0) mod A213999(n-1,n-1) = 0, and T(n,k+1) mod A213999(n,k) = 0 for 0 <= k <= n-1 (computed and verified for rows = 0..2000).
%F A347563 T(n,k) = Product_{p<=n} p^u_p, where u_p = i_max - Sum_{i=1..i_max} v_p(i) = Sum_{i=1..i_max} NOT(v_p(i)), with v_p(i) = floor(n/p^i) - floor(k/p^i) - floor((n-k)/p^i) = {0, 1} and i_max = floor(log(n)/log(p)) (using Erdős's method).
%e A347563 T(7,3) = 12. Triangle T(n,k) begins:
%e A347563      0;
%e A347563      1,   1;
%e A347563      2,   1,  2;
%e A347563      6,   2,  2,  6;
%e A347563     12,   3,  2,  3, 12;
%e A347563     60,  12,  6,  6, 12, 60;
%e A347563     60,  10,  4,  3,  4, 10, 60;
%e A347563    420,  60, 20, 12, 12, 20, 60, 420;
%e A347563    840, 105, 30, 15, 12, 15, 30, 105, 840;
%e A347563   2520, 280, 70, 30, 20, 20, 30,  70, 280, 2520;
%t A347563 Flatten[Table[(LCM@@Range(1,n))/Binomial[n, k], {n, 0, 11}, {k, 0, n}]]
%o A347563 (PARI) row(n) = vector(n+1, k, k--; lcm([1..n])/binomial(n,k)); \\ _Michel Marcus_, Sep 13 2021
%Y A347563 Cf. A003418, A007318, A213999.
%K A347563 nonn,tabl,easy
%O A347563 0,4
%A A347563 _Gary Waters_, Sep 06 2021