A080300 Global ranking function for totally balanced binary sequences.

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 5, 0, 0, 0, 0, 0, 6, 0, 7, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Antti Karttunen, Feb 21 2003

Keywords

• nonn

Comments

Note: the next nonzero value occurs at a(170)=9, as 170 = 10101010 is the lexicographically earliest totally balanced binary sequence of length 2*4.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..992
• A. Karttunen, Catalan ranking and unranking functions, OEIS Wiki.
• Various authors, Source code for Catalan ranking and unranking functions (in various programming languages), OEIS Wiki.

Crossrefs

Inverse function of A014486, i.e. a(A014486(n)) = n for all n. Cf. A080116, A215406, A213704, A209640.

Programs

Maple

A080300 := n -> A080116(n)*A215406(n); # Untested (as of Aug 19 2012)
A080300 := n -> `if`((0 = n) or (0 = A080116(n)),0, A014137(((A000523(n)+1)/2)-1)+A080301(n));

Mathematica

A080116[n_] := Module[{lev = 0, c = n}, While[c > 0, lev = lev + (-1)^c; c = Floor[c/2]; If[lev<0, Return[0]]]; If[lev>0, Return[0], Return[1]]];
A215406[n_] := Module[{m, d, a, y, t, x, u, v}, m = Quotient[Length[d = IntegerDigits[n, 2]], 2]; a = FromDigits[Reverse[d], 2]; y = 0; t = 1; For[x = 0, x <= 2*m - 2, x++, If[Mod[a, 2] == 1, y++, u = 2*m - x; v = m - Quotient[x + y, 2] - 1; t = t - Binomial[u - 1, v - 1] + Binomial[u - 1, v]; y--]; a = Quotient[a, 2]]; (1 - I*Sqrt[3])/2 - 4^(m + 1)*Gamma[m + 3/2]*Hypergeometric2F1[1, m + 3/2, m + 3, 4]/(Sqrt[Pi]*Gamma[m + 3]) -t];
a[n_] := A080116[n]*A215406[n] // Simplify;
Table[a[n], {n, 0, 170}] (* Jean-François Alcover, Mar 05 2016 *)

Formula

a(n) = A080116(n)*A215406(n).

a(n) = 0 if n=0 or (A080116(n)=0), otherwise a(n) = A014137(((A000523(n)+1)/2)-1)+A080301(n)