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%I #9 Aug 10 2012 14:03:56
%S 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,
%T 684,2730,0,0,0,56,180,690,2732,10922,0,0,0,0,184,692,2738,10924,
%U 43690,0,0,0,0,202,696,2740,10930,43692,174762,0,0,0,0,204,714,2744,10932,43698,174764,699050,0
%N 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.
%C 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.
%C See the comments and pictures at A014486 for more information.
%H Antti Karttunen, <a href="/A213704/b213704.txt">Rows n=0..54 of triangle, flattened</a>
%H A. Karttunen et al., <a href="http://oeis.org/wiki/Ranking_and_unranking_functions">Ranking and unranking functions</a>, OEIS Wiki.
%H F. Ruskey, <a href="http://www.cs.uvic.ca/~ruskey/Publications/Thesis/Thesis.html">Algorithmic Solution of Two Combinatorial Problems</a>, Thesis, Department of Applied Physics and Information Science, University of Victoria, 1978.
%o (Scheme):
%o (define (A213704 n) (A213704bi (A002262 n) (A025581 n)))
%o (define (A213704bi row col) (cond ((zero? row) 0) ((>= col (A000108 row)) 0) (else (CatalanUnrank row col))))
%o (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))))))
%Y The leftmost column: A020988. For all n>1, A014486(n) = A213704bi(A072643(n),(n - A014137(A072643(n)-1))). Cf. A009766, A215406, A153250.
%K nonn,tabl
%O 0,3
%A _Antti Karttunen_, Aug 10 2012
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