

A014486


List of totally balanced sequences of 2n binary digits written in base 10. Binary expansion of each term contains n 0's and n 1's and reading from left to right (the most significant to the least significant bit), the number of 0's never exceeds the number of 1's.


360



0, 2, 10, 12, 42, 44, 50, 52, 56, 170, 172, 178, 180, 184, 202, 204, 210, 212, 216, 226, 228, 232, 240, 682, 684, 690, 692, 696, 714, 716, 722, 724, 728, 738, 740, 744, 752, 810, 812, 818, 820, 824, 842, 844, 850, 852, 856, 866, 868, 872, 880
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OFFSET

0,2


COMMENTS

The binary DyckLanguage (A063171) in decimal representation.
These encode width 2n mountain ranges, rooted planar trees of n+1 vertices and n edges, planar planted trees with n nodes, rooted plane binary trees with n+1 leaves (2n edges, 2n+1 vertices, n internal nodes, the root included), Dyck words, binary bracketings, parenthesizations, noncrossing handshakes and partitions and many other combinatorial structures in Catalan family, enumerated by A000108.
Is Sum_{k=1..n} a(k)) / n^(5/2) bounded?  Benoit Cloitre, Aug 18 2002
This list is the intersection of A061854 and A031443.  Jason Kimberley, Jan 18 2013
The sequence does start at n = 0, since in the binary interpretation of the Dyck language (e.g., as parenthesizations where "1" stands for "(" and "0" stands for ")") having a(0) = 0 will do since it would stand for the empty string where the "0"s and "1"s are balanced (hence the parentheses are balanced).  Daniel Forgues, Feb 17 2013
It appears that for n>=1 this sequence can be obtained by concatenating the terms of the irregular array whose nth row length is A000108(n) and that is defined recursively by B(n,0) = A020988(n) and B(n,k) = B(n, k1) + D(n, k1) where D(x,y) = (2^(2*(A089309(B(x,y))1))1)*(2/3)+2^A007814(B(x,y)).  Raúl Mario Torres Silva and Michel Marcus, May 01 2020


LINKS

Franklin T. AdamsWatters, Table of n, a(n) for n = 0..2500
Jason Bell, Thomas Finn Lidbetter, Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
N. G. De Bruijn and B. J. M. Morselt, A note on plane trees, J. Combinatorial Theory 2 (1967), 2734.
Gennady Eremin, Dynamics of balanced parentheses, lexicographic series and Dyck polynomials, arXiv:1909.07675 [math.CO], 2019.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 320.
A. Karttunen, Catalan ranking and unranking functions, OEIS Wiki.
A. Karttunen, Illustration of 626 initial terms (up to size n=7) with various combinatorial interpretations of Catalan numbers encoded by this sequence.
A. Karttunen, a089408.c  C program for computing this sequence and many of the related automorphisms
A. Karttunen, Some notes on Catalan's Triangle
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020
D. L. Kreher and D. R. Stinson, Combinatorial Algorithms, Generation, Enumeration and Search, CRC Press, 1998.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
R. J. Mathar, Topologically Distinct Sets of Nonintersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
OEIS Wiki, Combinatorial interpretations of Catalan numbers
F. Ruskey, Algorithmic Solution of Two Combinatorial Problems
R. P. Stanley, Hipparchus, Plutarch, Schroeder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
R. P. Stanley, Exercises on Catalan and Related Numbers
Index entries for encodings of plane rooted trees (various subsets of this sequence).
Index entries for sequences related to parenthesizing
Index entries for signaturepermutations induced by Catalan automorphisms (permutations of natural numbers induced by various bijective operations acting on these structures)
Index entries for the sequences induced by list functions of Lisp (sequences induced by various other operations on these codes or the corresponding structures).


EXAMPLE

a(19) = 226_10 = 11100010_2 = A063171(19) as bracket expression: ( ( ( ) ) )( ) and as a binary tree, proceeding from left to right in depthfirst fashion, with 1's in binary expansion standing for internal (branching) nodes and 0's for leaves:
0 0
\ /
1 0 0 (0)
\ / \ /
1 1
\ /
1
Note that in this coding scheme the last leaf of the binary trees (here in parentheses) is implicit. This tree can be also converted to a particular Sexpression in languages like Lisp, Scheme and Prolog, if we interpret its internal nodes (1's) as cons cells with each leftward leaning branch being the "car" and the rightward leaning branch the "cdr" part of the pair, with the terminal nodes (0's) being ()'s (NILs). Thus we have (cons (cons (cons () ()) ()) (cons () ())) = '( ( ( () . () ) . () ) . ( () . () ) ) = (((())) ()) i.e., the same bracket expression as above, but surrounded by extra parentheses. This mapping is performed by the Scheme function A014486>parenthesization given below.


MAPLE

# Maple procedure CatalanUnrank is adapted from the algorithm 3.24 of the CAGES book and the Scheme function CatalanUnrank from Ruskey's thesis. See the a089408.c program for the corresponding Cprocedures.
CatalanSequences := proc(upto_n) local n, a, r; a := []; for n from 0 to upto_n do for r from 0 to (binomial(2*n, n)/(n+1))1 do a := [op(a), CatalanUnrank(n, r)]; od; od; return a; end;
CatalanUnrank := proc(n, rr) local r, x, y, lo, m, a; r := (binomial(2*n, n)/(n+1))(rr+1); y := 0; lo := 0; a := 0; for x from 1 to 2*n do m := Mn(n, x, y+1); if(r <= lo+m1) then y := y+1; a := 2*a + 1; else lo := lo+m; y := y1; a := 2*a; fi; od; return a; end;
Mn := (n, x, y) > binomial(2*nx, n((x+y)/2))  binomial(2*nx, n1((x+y)/2));


MATHEMATICA

cat[ n_ ] := (2 n)!/n!/(n+1)!; b2d[li_List] := Fold[2#1+#2&, 0, li]; d2b[n_Integer] := IntegerDigits[n, 2] tree[n_] := Join[Table[1, {i, 1, n}], Table[0, {i, 1, n}]]
nexttree[t_] := Flatten[Reverse[t]/. {a___, 0, 0, 1, b___}:> Reverse[{Sort[{a, 0}]//Reverse, 1, 0, b}]]
wood[ n_ /; n<8 ] := NestList[ nexttree, tree[ n ], cat[ n ]1 ]
Table[ Reverse[ b2d/@wood[ j ] ], {j, 0, 6} ]//Flatten
tbQ[n_]:=Module[{idn2=IntegerDigits[n, 2]}, Count[idn2, 1]==Length[idn2]/2&&Min[Accumulate[idn2/.{0>1}]]>=0]; Join[{0}, Select[Range[900], tbQ]] (* Harvey P. Dale, Jul 04 2013 *)
balancedQ[0] = True; balancedQ[n_] := Module[{s = 0}, Do[s += If[b == 1, 1, 1]; If[s < 0, Return[False]], {b, IntegerDigits[n, 2]}]; Return[s == 0] ]; A014486 = FromDigits /@ IntegerDigits[Select[Range[0, 1000], balancedQ ]] (* JeanFrançois Alcover, Mar 05 2016 *)
A014486Q[0] = True; A014486Q[n_] := Catch[Fold[If[# < 0, Throw[False], If[#2 == 0, #  1, # + 1]] &, 0, IntegerDigits[n, 2]] == 0]; Select[Range[0, 880], A014486Q] (* JungHwan Min, Dec 11 2016 *)


PROG

(MIT Scheme) (define (A014486 n) (let ((w/2 (A072643 n))) (CatalanUnrank w/2 (if (zero? n) 0 ( n (A014137 (1+ w/2)))))))
(Here 'm' is the row on A009766 and 'y' is the position on row 'm' of A009766, both >= 0. The resulting totally balanced binary string is computed into variable 'a'): (define (CatalanUnrank size rank) (let loop ((a 0) (m (1+ size)) (y size) (rank rank) (c (A009766 (1+ size) size))) (if (negative? m) a (if (>= rank c) (loop (1+ (* 2 a)) m (1+ y) ( rank c) (A009766 m (1+ y))) (loop (* 2 a) (1+ m) y rank (A009766 (1+ m) y))))))
(This converts the totally balanced binary string 'n' into the corresponding Sexpression:) (define (A014486>parenthesization n) (let loop ((n n) (stack (list (list)))) (cond ((zero? n) (car stack)) ((zero? (modulo n 2)) (loop (floor>exact (/ n 2)) (cons (list) stack))) (else (loop (floor>exact (/ n 2)) (cons2top! stack))))))
(define (cons2top! stack) (let ((excdr (cdr stack))) (setcdr! stack (car excdr)) (setcar! excdr stack) excdr))
(PARI) isA014486(n)=my(v=binary(n), t=0); for(i=1, #v, t+=if(v[i], 1, 1); if(t<0, return(0))); t==0 \\ Charles R Greathouse IV, Jun 10 2011
(Sage)
def is_A014486(n) :
B = bin(n)[2::] if n != 0 else 0
s = 0
for b in B :
s += 1 if b=='1' else 1
if 0 > s : return False
return 0 == s
def A014486_list(n): return [k for k in (1..n) if is_A014486(k) ]
A014486_list(888) # Peter Luschny, Aug 10 2012


CROSSREFS

Characteristic function: A080116. Inverse function: A080300.
The terms of binary width 2n are counted by A000108[n]. Subset of A036990. Number of peaks in each mountain (number of leaves in rooted plane general trees): A057514. Number of trailing zeros in the binary expansion: A080237. First differences: A085192.
Cf. also A009766, A014137, A071156, A072643, A079436, A085184, A213704.
Cf. A020988, A089309, A007814.
Sequence in context: A186630 A154391 A035928 * A166751 A216649 A071162
Adjacent sequences: A014483 A014484 A014485 * A014487 A014488 A014489


KEYWORD

nonn,nice,easy,base


AUTHOR

Wouter Meeussen


EXTENSIONS

Additional comments from Antti Karttunen, Aug 11 2000 and May 25 2004
Added a(0)=0 (which had been removed in June 2011), Joerg Arndt, Feb 27 2013


STATUS

approved



