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

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, 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, 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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

T(h,k) = number of periodic binary words of k 1's and h-k 0's.

Examples

			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;
1,0,0,0,0,0,0,1;
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;
		

Crossrefs

Cf. A007318. Different from A053200.

Programs

Maple

A050186 := proc(n,k)
        if n = 0 then
                1;
        else
        add (numtheory[mobius](d)*binomial(n/d,k/d),d =numtheory[divisors](igcd(n,k))) ;
        end if;
end proc:
A050870 := proc(n,k)
        binomial(n,k)-A050186(n,k) ;
end proc:
seq(seq(A050870(n,k),k=0..n),n=0..20) ; # R. J. Mathar, Sep 24 2011

Mathematica

T[n_, k_] := Binomial[n, k] - If[n == 0, 1, Sum[MoebiusMu[d] Binomial[n/d, k/d], {d, Divisors[GCD[n, k]]}]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}] (* Jean-François Alcover, Jul 01 2019 *)

Extensions

Edited by N. J. A. Sloane, Aug 29 2008