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

0,11

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.

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)

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 *)

Prog

(See the Source code... page at OEIS Wiki! Please add your code there, if possible.)

Crossrefs

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

Sequence in context: A013371 A013372 A209640 * A116864 A255308 A079302

Adjacent sequences: A080297 A080298 A080299 * A080301 A080302 A080303

Keyword

nonn

Author

Antti Karttunen, Feb 21 2003

Status

approved