0,3

There are two schools of thought about the index for the first term. I prefer the indexing a(0) = a(1) = 1, a(2) = 3, a(3) = 11, etc.

a(n) = number of ways to insert parentheses in a string of n symbols. The parentheses must be balanced but there is no restriction on the number of pairs of parentheses. The number of letters inside a pair of parentheses must be at least 2. Parentheses enclosing the whole string are ignored.

Also length of list produced by a variant of the Catalan producing iteration: replace each integer k by the list 0,1,..,k,k+1,k,...,1,0 and get the length a(n) of the resulting (flattened) list after n iterations. - Wouter Meeussen (wouter.meeussen(AT)pandora.be), Nov 11 2001

Stanley gives several other interpretations for these numbers.

Number of Schroeder paths of semilength n-1 (i.e. lattice paths from (0,0) to (2n-2,0), with steps H=(2,0), U=(1,1) and D(1,-1) and not going below the x-axis) with no peaks at level 1. Example: a(3)=3 because among the six Schroeder paths of semilength two HH, UHD, UUDD, HUD, UDH and UDUD, only the first three have no peaks at level 1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 27 2003

a(n+1)=number of Dyck n-paths in which the interior vertices of the ascents are colored white or black. - David Callan (callan(AT)stat.wisc.edu), Mar 14 2004

Number of possible schedules for n time slots in the first-come first-served (FCFS) printer policy.

Also row sums of A086810, A033282 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), May 09 2004

a(n+1) = number of pairs (u,v) of same-length compositions of n, 0's allowed in u but not in v and u dominates v (meaning u_1 >= v_1, u_1 + u_2 >= v_1 + v_2 and so on). For example, with n=2, a(3) counts (2,2), (1+1,1+1), (2+0,1+1). - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005

The big Schroeder number (A006318) is the number of Schroeder paths from (0,0) to (n,n) (subdiagonal paths with steps (1,0) (0,1) and (1,1)). These paths fall in two classes: those with steps on the main diagonal and those without. These two classes are equinumerous and the number of paths in either class is the little Schroeder number a(n) (half the big Schroeder number). - Marcelo Aguiar (maguiar(AT)math.tamu.edu), Oct 14 2005

With offset 0, a(n) = number of (colored) Motzkin (n-1)-paths with each upstep U getting one of 2 colors and each flatstep F getting one of 3 colors. Example. With their colors immediately following upsteps/flatsteps, a(2) = 3 counts F1, F2, F3 and a(3)=11 counts U1D, U2D, F1F1, F1F2, F1F3, F2F1, F2F2, F2F3, F3F1, F3F2, F3F3. - David Callan (callan(AT)stat.wisc.edu), Aug 16 2006

Triangle A144156 has row sums = A006318 with left border A001003. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]

Shifts left when INVERT transform applied twice. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Apr 01 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.

D. Arques and A. Giorgetti, Une bijection geometrique entre une famille d'hypercartes et une famille de polygones enumerees par la serie de Schroeder, Discrete Math., 217 (2000), 17-32.

P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 57.

E. Deutsch, A bijective proof of an equation linking the Schroeder numbers, large and small, Discrete Math., 241 (2001), 235-240.

I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6.

D. Foata and D. Zeilberger, A classic proof of a recurrence for a very classical sequence, J. Comb Thy A 80 380-384 1997.

D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6, Problem 66.

G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.

J. S. Lew, Polynomial enumeration of multidimensional lattices, Math. Systems Theory, 12 (1979), 253-270.

T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 168.

E. Schroeder, Vier combinatorische Probleme, Zeit. f. Math. Phys., 15 (1870), 361-376.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 178.

H. N. V. Temperley and D. G. Rogers, A note on Baxter's generalization of the Temperley-Lieb operators, pp. 324-328 of Combinatorial Mathematics (Canberra, 1977), Lect. Notes Math. 686, 1978.

I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 198.

C. Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.

D. Gouyou-Beauchamps and B. Vauquelin, Deux proprietes combinatoires des nombres de Schroeder, Theor. Inform. Appl., 22 (1988), 361-388.

M. Klazar, On numbers of Davenport-Schinzel sequences, Discr. Math., 185 (1998), 77-87.

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Wen-jin Woan, A Recursive Relation for Weighted Motzkin Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.6.

Wen-jin Woan, A Relation Between Restricted and Unrestricted Weighted Motzkin Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.7.

T. D. Noe, Table of n, a(n) for n=0..199

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Marcelo Aguiar and Walter Moreira, Combinatorics of the free Baxter algebra, see Corollary 3.3.iii.

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.

E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, ECO method and hill-free generalized Motzkin paths

H. Bottomley, Illustration of initial terms

F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).

W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns

S.-P. Eu and T.-S. Fu, A simple proof of the Aztec diamond problem

D. Foata and D. Zeilberger, [math/9805015] A Classic Proof of a Recurrence for a Very Classical Sequence

D. Foata and D. Zeilberger, A classic proof of a recurrence for a very classical sequence

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 42

D. Merlini, R. Sprugnoli and M. C. Verri, Waiting patterns for a printer, FUN with algorithm'01, Isola d'Elba, 2001.

P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1.

E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.

N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences

L. M. Smiley, Variants of Schroeder Dissections

R. P. Stanley, Hipparchus, Plutarch, Schr"oder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to parenthesizing

N. J. A. Sloane, Transforms [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Apr 01 2009]

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 69

a(n) = 3*a(n-1) + 2*A065096(n-2) (n>2). If g(x) = 1 + 3x + 11x^2 + 45x^3 + ... + a(n)*x^n + ..., then g(x) = 1 + 3(x*g(x)) + 2(x*g(x))^2, g(x)^2 = 1 + 6x + 31x^2 + 156x^3 + ... + A065096(n)*x^n + ... - Paul D. Hanna (pauldhanna(AT)juno.com), Jun 10 2002

a(n+1) = -a(n) + 2*Sum_{k=1..n} a(k)*a(n+1-k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 27 2004

The Hankel transform of this sequence gives A006125 = 1, 1, 2, 8, 64, 1024, ...; example : det([1, 1, 3, 11; 1, 3, 11, 45; 3, 11, 45, 197; 11, 45, 197, 903]) = 2^6 = 64 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 02 2004

a(n+1)=Sum_{k, 0, (n-1)/2} 2^k 3^(n-1-2k) binomial(n-1, 2k) CatalanNumber(k). This formula counts colored Dyck paths by the same parameter by which Touchard's identity counts ordinary Dyck paths: number of DDUs (U=up step, D=down step). See also Gouyou-Beauchamps reference. - David Callan (callan(AT)stat.wisc.edu), Mar 14 2004

a(n)=(1/(n+1))sum{k=0..n, C(n+1, k)C(2n-k, n)(-1)^k*2^(n-k)} [with offset 0]; a(n)=(1/(n+1))sum{k=0..n, C(n+1, k+1)C(n+k, k)(-1)^(n-k)*2^k} [with offset 0]; a(n)=sum{k=0..n, (1/(k+1))*C(n, k)C(n+k, k)(-1)^(n-k)*2^k} [with offset 0]; a(n)=sum{k=0..n, A088617(n, k)*(-1)^(n-k)*2^k} [with offset 0]; - Paul Barry (pbarry(AT)wit.ie), May 24 2005

(n+1)*a(n)= (6*n-3)* a(n-1) -(n-2)*a(n-2) if n>1. a(0)= a(1)= 1.

G.f.: (1 + x - sqrt(1 - 6*x + x^2) )/(4*x) = 2/(1 + x + sqrt(1 - 6*x + x^2)) .

E.g.f. of a(n+1)= exp(3*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 31 2004

Reversion of (x-2*x^2)/(1-x) is g.f. offset 1.

For n>=1, a(n)=sum(k=0, n, 2^k*N(n, k)) where N(n, k) =1/n*C(n, k)*C(n, k+1) are the Narayana numbers (A001263) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 10 2003. This formula counts colored Dyck paths by number of peaks, which is easy to see because the Narayana numbers count Dyck paths by number of peaks and the number of peaks determines the number of interior ascent vertices.

a(n) = Sum_{k=0..n} A088617(n, k)*2^k*(-1)^(n-k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 21 2004

For n > 0, a(n) = 1/(n+1) * sum_{k = 0 .. n-1} binomial(2*n-k, n)*binomial(n-1, k). This formula counts colored Dyck paths (as above) by number of white vertices. - David Callan (callan(AT)stat.wisc.edu), Mar 14 2004

a(n-1)=(diff(((1-x)/(1-2*x))^n, x$(n-1)))/n!|_{x=0}. (For a proof see the comment on the unsigned row sums of triangle A111785.)

a(n)= (1/n)*sum(binomial(n, k)*binomial(n+k, k-1), k=1..n) = hypergeom([1-n, n+2], [2], -1), n>=1. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 12 2005.

a(m+n+1) = Sum_{k, k>=0} A110440(m, k)*A110440(n, k)*2^k = A110440(m+n, 0) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 14 2005

Sum over partitions formula (reference Schroeder paper p. 362, eq. (1) II). Number the partitions of n according to Abramowitz-Stegun p. 831-2 (see reference under A105805) with k=1..p(n)= A000041(n). For n>=1: a(n-1)=sum(A048996(n,k)*a(1)^e(k, 1)*a(1)^e(k, 2)*...*a(n-2)^e(k, n-1), k=2..p(n)) if the k-th partition of n in the mentioned order is written as (1^e(k, 1), 2^e(k, 2), ..., (n-1)e(k, n-1)). Note that the first (k=1) partition (n^1) has to be omitted. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 23 2005.

Starting (1, 3, 11, 45,...), = row sums of triangle A126216 = A001263 * [1, 2, 4, 8, 16,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2007

Contribution from Paul Barry (pbarry(AT)wit.ie), May 15 2009: (Start)

G.f.: 1/(1+x-2x/(1+x-2x/(1+x-2x/(1+x-2x/(1-.... (continued fraction).

G.f.: 1/(1-x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction).

G.f.: 1/(1-x-2x^2/(1-3x-2x^2/(1-3x-2x^2/(1-... (continued fraction). (End)

a(2) = 3: abc, a(bc), (ab)c; a(3) = 11: abcd, (ab)cd, a(bc)d, ab(cd), (ab)(cd), a(bcd), a(b(cd)), a((bc)d), (abc)d, (a(bc))d, ((ab)c)d.

Sum over partitions formula: a(3) = 2*a(0)*a(2) + 1*a(1)^2 + 3*(a(0)^2)*a(1) + 1*a(0)^4 = 6 + 1 + 3 + 1 = 11.

t1 := (1/(4*x))*(1+x-sqrt(1-6*x+x^2)); series(t1, x, 40);

BB:=(1+z-sqrt(1-6*z+z^2))/4: BBser:=series(BB, z=0, 26): seq(coeff(BBser, z, n), n=1..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 10 2007

seq (sum(binomial(n, j)*binomial(n+j, n-1), j=0..n)/(2*n), n=1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 10 2007

invtr:= proc(p) local b; b:= proc(n) option remember; local i; `if` (n<1, 1, add (b(n-i) *p(i-1), i=1..n+1)) end; end: a:='a': f:= (invtr@@2)(a): a:= proc (n) if n<0 then 1 else f(n-1) fi end: seq (a(n), n=0..30); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Apr 01 2009]

Table[Length[Flatten[Nest[ #/.a_Integer:> Join[Range[0, a+1], Range[a, 0, -1]]&, {0}, n]]], {n, 0, 10}]

Sch[ 0 ]=Sch[ 1 ]=1; Sch[ n_Integer ] := Sch[ n ]=(3(2n-1)Sch[ n-1 ]-(n-2)Sch[ n-2 ])/(n+1); Array[ Sch[ #-1 ]&, 20 ]

(PARI) {a(n)= if(n<1, n==0, sum(k=0, n, 2^k*binomial(n, k)* binomial(n, k-1) )/(2*n))} /* Michael Somos Mar 31 2007 */

(PARI) {a(n)= local(A); if(n<1, n==0, n--; A= x*O(x^n); n!*simplify(polcoeff( exp(3*x +A)* besseli(1, 2*x* quadgen(8) +A), n)))} /* Michael Somos Mar 31 2007 */

(PARI) {a(n)= if(n<0, 0, n++; polcoeff( serreverse( (x -2*x^2)/(1 -x) +x*O(x^n)), n))} /* Michael Somos Mar 31 2007 */

See A000108, A001190, A001699, A000081 for other ways to count parentheses. Cf. A000311, A010683, A065096.

Row sums of A033877.

Cf. A054726, A059435, A025240, A080243, A085403, A086456, A086616, A035011.

Right-hand column 1 of triangle A011117.

Convolution triangle A011117.

A006318(n)= 2*(n) if n>0.

A144156 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]

Sequence in context: A146086 A049177 A080243 this_sequence A151131 A151132 A151133

Adjacent sequences: A001000 A001001 A001002 this_sequence A001004 A001005 A001006

nonn,easy,nice,core

N. J. A. Sloane (njas(AT)research.att.com).