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A050870 T(h,k) = binomial(h,k) - A050186(h,k).

%I A050870 #18 Jul 01 2019 14:07:35
%S A050870 0,0,0,1,0,1,1,0,0,1,1,0,2,0,1,1,0,0,0,0,1,1,0,3,2,3,0,1,1,0,0,0,0,0,
%T A050870 0,1,1,0,4,0,6,0,4,0,1,1,0,0,3,0,0,3,0,0,1,1,0,5,0,10,2,10,0,5,0,1,1,
%U A050870 0,0,0,0,0,0,0,0,0,0,1,1,0,6,4,15,0,24,0,15,4,6,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,7,0,21,0,35,2,35,0,21,0,7,0,1,1,0
%N A050870 T(h,k) = binomial(h,k) - A050186(h,k).
%C A050870 T(h,k) = number of periodic binary words of k 1's and h-k 0's.
%e A050870 0;
%e A050870 0,0;
%e A050870 1,0,1;
%e A050870 1,0,0,1;
%e A050870 1,0,2,0,1;
%e A050870 1,0,0,0,0,1;
%e A050870 1,0,3,2,3,0,1;
%e A050870 1,0,0,0,0,0,0,1;
%e A050870 1,0,4,0,6,0,4,0,1;
%e A050870 1,0,0,3,0,0,3,0,0,1;
%e A050870 1,0,5,0,10,2,10,0,5,0,1;
%p A050870 A050186 := proc(n,k)
%p A050870         if n = 0 then
%p A050870                 1;
%p A050870         else
%p A050870         add (numtheory[mobius](d)*binomial(n/d,k/d),d =numtheory[divisors](igcd(n,k))) ;
%p A050870         end if;
%p A050870 end proc:
%p A050870 A050870 := proc(n,k)
%p A050870         binomial(n,k)-A050186(n,k) ;
%p A050870 end proc:
%p A050870 seq(seq(A050870(n,k),k=0..n),n=0..20) ; # _R. J. Mathar_, Sep 24 2011
%t A050870 T[n_, k_] := Binomial[n, k] - If[n == 0, 1, Sum[MoebiusMu[d] Binomial[n/d, k/d], {d, Divisors[GCD[n, k]]}]];
%t A050870 Table[T[n, k], {n, 0, 15}, {k, 0, n}] (* _Jean-François Alcover_, Jul 01 2019 *)
%Y A050870 Cf. A007318. Different from A053200.
%K A050870 nonn,tabl
%O A050870 0,13
%A A050870 _Clark Kimberling_
%E A050870 Edited by _N. J. A. Sloane_, Aug 29 2008