A080301 Local ranking function for totally balanced binary sequences: if n's binary expansion is totally balanced (A080116(n)=1), then a(n) is its zero-based position among A000108((A000523(n)+1)/2) lexicographically ordered totally balanced binary sequences of the same width, otherwise -1.

0, -1, 0, -1, -1, -1, -1, -1, -1, -1, 0, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, -1, 1, -1, -1, -1, -1, -1, 2, -1, 3, -1, -1, -1, 4, -1, -1, -1, -1

Offset: 0

Author

Antti Karttunen, Feb 21 2003

Keywords

• sign

Comments

Maple procedure CatalanRank is adapted from the algorithm 3.23 of the CAGES book.

Examples

			We have Cat(0)=1 totally balanced binary sequences of length 2*0: 0, thus a(0)=0, Cat(1)=1 of length 2*1: 10, thus a(2)=0, Cat(2)=2 of length 2*2: 1010 (= 10.) and 1100 (= 12.), thus a(10)=0 and a(12)=1, plus altogether Cat(3)=5 totally balanced binary sequences of length 2*3: 101010 (= 42), 101100 (= 44), 110010 (= 50), 110100 (= 52), 111000 (= 56), thus a(42)=0, a(44)=1, a(50)=2, a(52)=3 and a(56)=4. Et cetera.
		

Links

• D. L. Kreher and D. R. Stinson, Combinatorial Algorithms, Generation, Enumeration and Search, CRC Press, 1998.

Crossrefs

Used to compute A080300. Cf. A009766, A000523.

Programs

Maple

A080301 := n -> `if`(0 = A080116(n),-1,CatalanRank((A000523(n)+1)/2,n));
CatalanRank := proc(n,aa) local y,r,lo,a; a := aa; r := 0; y := -1; lo := 0; while (a > 0) do if(0 = (a mod 2)) then r := r+1; lo := lo + A009766(r,y); else y := y+1; fi; a := floor(a/2); od; RETURN((binomial(2*n,n)/(n+1))-(lo+1)); end;