A215406 -id:A215406 - cp's OEIS frontend

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

Antti Karttunen, Feb 21 2003

nonn

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.

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.

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

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));

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 *)

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)

A080119 Positions of A080118 in A014486.

Original entry on oeis.org

1, 2, 7, 33, 81, 74395, 8369196, 215802898, 414859094165, 520973680640109, 4064761999842441067, 517978450857911919447, 4255027826896017770661, 5222501054779098990032001033, 718000720375918750838217734094612383

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

A080119 := n -> CatalanRankGlobal(A080118(n));

def A080119(n) : return A215406(A080118(n))
[A080119(n) for n in (1..15)] # Peter Luschny, Aug 10 2012

a(n) = A215406(A080118(n)). - Peter Luschny, Aug 10 2012

A213704 Catalan Unranking function U(size,rank) for totally balanced binary strings (converted to decimal). Each row 'size' of an array lists all A000108(size) such items in standard lexicographic order, followed by an infinite number of zeros.

Original entry on oeis.org

0, 0, 2, 0, 0, 10, 0, 0, 12, 42, 0, 0, 0, 44, 170, 0, 0, 0, 50, 172, 682, 0, 0, 0, 52, 178, 684, 2730, 0, 0, 0, 56, 180, 690, 2732, 10922, 0, 0, 0, 0, 184, 692, 2738, 10924, 43690, 0, 0, 0, 0, 202, 696, 2740, 10930, 43692, 174762, 0, 0, 0, 0, 204, 714, 2744, 10932, 43698, 174764, 699050, 0

Offset: 0

Antti Karttunen, Aug 10 2012

The Scheme-function CatalanUnrank has been adapted from Frank Ruskey's thesis. This gives essentially the same information as A014486 which can be obtained from this array by concatenating all A000108(s) nonzero terms from the beginning of each row s to one sequence.

See the comments and pictures at A014486 for more information.

Antti Karttunen, Rows n=0..54 of triangle, flattened
Antti Karttunen et al., Ranking and unranking functions, OEIS Wiki.
Frank Ruskey, Algorithmic Solution of Two Combinatorial Problems, Thesis, Department of Applied Physics and Information Science, University of Victoria, 1978.

The leftmost column: A020988. For all n>1, A014486(n) = A213704bi(A072643(n),(n - A014137(A072643(n)-1))). Cf. A009766, A215406, A153250.

(define (A213704 n) (A213704bi (A002262 n) (A025581 n)))
(define (A213704bi row col) (cond ((zero? row) 0) ((>= col (A000108 row)) 0) (else (CatalanUnrank row col))))
(define (CatalanUnrank size rank) (let loop ((a 0) (m (-1+ size)) (y size) (rank rank) (c (A009766tr (-1+ size) size))) (if (negative? m) a (if (>= rank c) (loop (1+ (* 2 a)) m (-1+ y) (- rank c) (A009766tr m (-1+ y))) (loop (* 2 a) (-1+ m) y rank (A009766tr (-1+ m) y))))))

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A080300 Global ranking function for totally balanced binary sequences.

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A080119 Positions of A080118 in A014486.

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A213704 Catalan Unranking function U(size,rank) for totally balanced binary strings (converted to decimal). Each row 'size' of an array lists all A000108(size) such items in standard lexicographic order, followed by an infinite number of zeros.

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