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A001006 Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.
(Formerly M1184 N0456)
1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829 (list; graph; refs; listen; history; text; internal format)



Number of 4321-, (3412,2413)-, (3412,3142)- and 3412-avoiding involutions in S_n.

Number of sequences of length n-1 consisting of positive integers such that the opening and ending elements are 1 or 2 and the absolute difference between any 2 consecutive elements is 0 or 1. - Jon Perry, Sep 04 2003

Also number of Motzkin n-paths: paths from (0,0) to (n,0) in an n X n grid using only steps U = (1,1), F = (1,0) and D = (1,-1). - David Callan, Jul 15 2004

Number of Dyck n-paths with no UUU. (Given such a Dyck n-path, change each UUD to U, then change each remaining UD to F. This is a bijection to Motzkin n-paths. Example with n=5: U U D U D U U D D D -> U F U D D.) - David Callan, Jul 15 2004

Number of Dyck (n+1)-paths with no UDU. (Given such a Dyck (n+1)-path, mark each U that is followed by a D and each D that is not followed by a U. Then change each unmarked U whose matching D is marked to an F. Lastly, delete all the marked steps. This is a bijection to Motzkin n-paths. Example with n=6 and marked steps in small type: U U u d D U U u d d d D u d -> U U u d D F F u d d d D u d -> U U D F F D.) - David Callan, Jul 15 2004

a(n) is the number of strings of length 2n from the following recursively defined set: L contains the empty string and, for any strings a and b in L, we also find (ab) in L. The first few elements of L are e, (), (()), ((())), (()()), (((()))), ((()())), ((())()), (()(())) and so on. This proves that a(n) is less than or equal to C(n), the n-th Catalan number. - Saul Schleimer (saulsch(AT)math.rutgers.edu), Feb 23 2006

a(n) = number of Dyck n-paths all of whose valleys have even x-coordinate (when path starts at origin). For example, T(4,2)=3 counts UDUDUUDD, UDUUDDUD, UUDDUDUD. Given such a path, split it into n subpaths of length 2 and transform UU->U, DD->D, UD->F (there will be no DUs for that would entail a valley with odd x-coordinate). This is a bijection to Motzkin n-paths. - David Callan, Jun 07 2006

Also the number of standard Young tableaux of height <= 3. - Mike Zabrocki, Mar 24 2007

a(n) is the number of RNA shapes of size 2n+2. RNA Shapes are essentially Dyck words without "directly nested" motifs of the form A[[B]]C, for A, B and C Dyck words. The first RNA Shapes are []; [][]; [][][], [[][]]; [][][][], [][[][]], [[][][]], [[][]][]; ... - Yann Ponty (ponty(AT)lri.fr), May 30 2007

Equals right and left borders and row sums of triangle A144218 with offset variations. - Gary W. Adamson, Sep 14 2008

The sequence is self-generated from top row A going to the left starting (1,1) and bottom row = B, the same sequence but starting (0,1) and going to the right. Take dot product of A and B and add the result to n-th term of A to get the (n+1)-th term of A. Example: a(5) = 21 as follows: Take dot product of A = (9, 4, 2, 1, 1) and (0, 1, 1, 2, 4) = (0, + 4 + 2 + 2 + 4) = 12; which is added to 9 = 21. - Gary W. Adamson, Oct 27 2008

Equals A005773 / A005773 shifted (i.e., (1,2,5,13,35,96,...) / (1,1,2,5,13,35,96,...)). - Gary W. Adamson, Dec 21 2008

Starting with offset 1 = iterates of M * [1,1,0,0,0,...], where M = a tridiagonal matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. - Gary W. Adamson, Jan 07 2009

a(n) is the number of involutions of {1,2,...,n} having genus 0. The genus g(p) of a permutation p of {1,2,...,n} is defined by g(p)=(1/2)[n+1-z(p)-z(cp')], where p' is the inverse permutation of p, c = 234...n1 = (1,2,...,n), and z(q) is the number of cycles of the permutation q. Example: a(4)=9; indeed, p=3412=(13)(24) is the only involution of {1,2,3,4} with genus >0. This follows easily from the fact that a permutation p of {1,2,...,n} has genus 0 if and only if the cycle decomposition of p gives a noncrossing partition of {1,2,...,n} and each cycle of p is increasing (see Lemma 2.1 of the Dulucq-Simion reference). [Also, redundantly, for p=3412=(13)(24) we have cp'=2341*3412=4123=(1432) and so g(p)=(1/2)(4+1-2-1)=1.] - Emeric Deutsch, May 29 2010

Let w(i,j,n) denote walks in N^2 which satisfy the multivariate recurrence

w(i,j,n) = w(i, j + 1, n - 1) + w(i - 1, j, n - 1) + w(i + 1, j - 1, n - 1) with boundary conditions w(0,0,0) = 1 and w(i,j,n) = 0 if i or j or n is < 0. Then a(n) = Sum{i = 0..n,j = 0..n} w(i,j,n) is the number of such walks of length n. - Peter Luschny, May 21 2011

a(n)/a(n-1) tends to 3.0 as Lim N->inf: (1+2*Cos 2Pi/N) relating to longest odd N regular polygon diagonals, by way of example, N=7: Using the tridiagonal generator [Cf. comment of Jan 07 2009], for polygon N=7, we extract an (N-1)/2 = 3 X 3 matrix, [0,1,0; 1,1,1; 0,1,1] with an e-val of 2.24697...; the longest Heptagon diagonal with edge = 1. As N tends to infinity, the diagonal lengths tend to 3.0, the convergent of the sequence. - Gary W. Adamson, Jun 08 2011

Number of (n+1)-length permutations avoiding the pattern 132 and the dotted pattern 23\dot{1}. - Jean-Luc Baril, Mar 07 2012

Number of n-length words w over alphabet {a,b,c} such that for every prefix z of w we have #(z,a) >= #(z,b) >= #(z,c), where #(z,x) counts the letters x in word z.  The a(4) = 9 words are: aaaa, aaab, aaba, abaa, aabb, abab, aabc, abac, abca. - Alois P. Heinz, May 26 2012

Number of length-n restricted growth strings (RGS) [r(1), r(2), ..., r(n)] such that r(1)=1, r(k)<=k, and r(k)!=r(k-1); for example, the 9 RGS for n=4 are 1010, 1012, 1201, 1210, 1212, 1230, 1231, 1232, 1234. - Joerg Arndt, Apr 16 2013

Number of length-n restricted growth strings (RGS) [r(1), r(2), ..., r(n)] such that r(1)=0, r(k)<=k and r(k)-r(k-1) != 1; for example, the 9 RGS for n=4 are 0000, 0002, 0003, 0004, 0022, 0024, 0033, 0222, 0224. - Joerg Arndt, Apr 17 2013

Number of (4231,5276143)-avoiding involutions in S_n. - Alexander Burstein, Mar 05 2014

a(n) is the number of increasing unary-binary trees with n nodes who have an associated permutation avoids 132. For more information about unary-binary trees with associated permutations, see A245888. - Manda Riehl, Aug 07 2014

a(n) is the number of involutions on [n] avoiding the single pattern p, where p is any one of the 8 (classical) patterns 1234, 1243, 1432, 2134, 2143, 3214, 3412, 4321. Also, number of (3412,2413)-, (3412,3142)-, (3412,2413,3142)-avoiding involutions on [n] because each of these 3 sets actually coincides with the 3412-avoiding involutions on [n]. This is a complete list of the 8 singles, 2 pairs, and 1 triple of 4-letter classical patterns whose involution avoiders  are counted by the Motzkin numbers. (See Barnabei et al 2011 reference.) - David Callan, Aug 27 2014

A series created using 2*a(n)+ a(n+1) has Hankel transform of F(2n), offset 3, F being the Fibonacci bisection, A001906 (Empirical observation). - Tony Foster III, Jul 28 2016

A series created using 2*a(n) + 3*a(n+1) + a(n+2) gives the Hankel transform of Sum(k*Fibonacci(2*k), k=0..n), offset 3, A197649 (Empirical observation). - Tony Foster III, Jul 28 2016


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A. Sapounakis and P. Tsikouras, On k-colored Motzkin words, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.5.

E. Schröder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376. [Annotated scanned copy]

N. J. A. Sloane, Illustration of initial terms

N. J. A. Sloane, Classic Sequences

N. J. A. Sloane, An Application of the OEIS (Vugraph from a talk about the OEIS)

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers [Corrected annotated scanned copy]

R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.

Yidong Sun and Fei Ma, Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths, arXiv preprint arXiv:1305.2015 [math.CO], 2013.

Zhi-Wei Sun, Conjectures involving combinatorial sequences, arXiv preprint arXiv:1208.2683 [math.CO], 2012.

Paul Tarau, On logic programming representations of lambda terms: de Bruijn indices, compression, type inference, combinatorial generation, normalization, 2015.

P. Tarau, A Logic Programming Playground for Lambda Terms, Combinators, Types and Tree-based Arithmetic Computations, arXiv preprint arXiv:1507.06944, 2015

Yi Wang and Bao-Xuan Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, arXiv preprint arXiv:1303.5595 [math.CO], 2013.

Eric Weisstein's World of Mathematics, Motzkin Number

W.-J. Woan, Hankel Matrices and Lattice Paths, J. Integer Sequences, 4 (2001), #01.1.2.

J. Y. X. Yang, M. X. X. Zhong, R. D. P. Zhou, On the Enumeration of (s, s+ 1, s+2)-Core Partitions, arXiv preprint arXiv:1406.2583 [math.CO], 2014.

Huan Xiong, The number of simultaneous core partitions, arXiv preprint arXiv:1409.7038 [math.CO], 2014.

Yan X Zhang, Four Variations on Graded Posets, arXiv preprint arXiv:1508.00318, 2015

Index entries for "core" sequences


G.f.: A(x) = ( 1 - x - (1-2*x-3*x^2)^(1/2) ) / (2*x^2).

G.f. A(x) satisfies A(x) = 1 + x*A(x) + x^2*A(x)^2.

G.f. F(x)/x where F(x) is the reversion of x/(1+x+x^2). - Joerg Arndt, Oct 23 2012

a(n) = (-1/2) Sum_(-3)^i C(1/2, i) C(1/2, j); i+j=n+2, i >= 0, j >= 0.

a(n) = (3/2)^(n+2) * Sum_{k >= 1} 3^(-k) * Catalan(k-1) * binomial(k, n+2-k). [Doslic et al.]

a(n) ~ 3^(n+1)sqrt(3)[1+1/(16n)]/[(2n+3)sqrt((n+2)Pi)]. [Barcucci, Pinzani and Sprugnoli]

Lim_{n->infinity} a(n)/a(n-1) = 3. [Aigner]

a(n+2) - a(n+1) = a(0)*a(n) + a(1)*a(n-1) + ... + a(n)*a(0). [Bernhart]

a(n) = (1/(n+1)) * Sum_{i} (n+1)!/(i!*(i+1)!*(n-2*i)!). [Bernhart]

a(n) = sum((-1)^(n-k)*binomial(n, k)*A000108(k+1), k=0..n). a(n) = sum(binomial(n+1, k)*binomial(n+1-k, k-1), k=0..ceil((n+1)/2))/(n+1); (n+2)a(n) = (2n+1)a(n-1)+(3n-3)a(n-2). - Len Smiley

a(n) = sum{ k=0..n, C(n, 2k)*A000108(k) }. - Paul Barry, Jul 18 2003

E.g.f.: exp(x)*BesselI(1, 2*x)/x. - Vladeta Jovovic, Aug 20 2003

a(n) = A005043(n) + A005043(n+1).

The Hankel transform of this sequence gives A000012 = [1, 1, 1, 1, 1, 1, ...]. E.g., Det([1, 1, 2, 4; 1, 2, 4, 9; 2, 4, 9, 21; 4, 9, 21, 51]) = 1. - Philippe Deléham, Feb 23 2004

a(m+n) = Sum_{k>=0} A064189(m, k)*A064189(n, k). - Philippe Deléham, Mar 05 2004

a(n) = sum((-1)^j*binomial(n+1, j)*binomial(2n-3j, n), j=0..floor(n/3))/(n+1). - Emeric Deutsch, Mar 13 2004

a(n) = A086615(n)-A086615(n-1) (n>=1). - Emeric Deutsch, Jul 12 2004

G.f.: A(x)=(1-y+y^2)/(1-y)^2 where (1+x)*(y^2-y)+x=0; A(x)=4*(1+x)/(1+x+sqrt(1-2*x-3*x^2))^2; a(n)=(3/4)*(1/2)^n*sum(k=0..2*n, 3^(n-k)*C(k)*C(k+1, n+1-k) ) + 0^n/4 [after Doslic et al.]. - Paul Barry, Feb 22 2005

G.f.: c(x^2/(1-x)^2)/(1-x), c(x) the g.f. of A000108. - Paul Barry, May 31 2006

Asymptotic formula : a(n) ~ sqrt(3/4/Pi)*3^(n+1)/n^(3/2). - Benoit Cloitre, Jan 25 2007

a(n) = A007971(n+2)/2. - Zerinvary Lajos, Feb 28 2007

a(n)=(1/(2*pi))*int(x^n*sqrt((3-x)*(1+x)),x,-1,3) is the moment representation. - Paul Barry, Sep 10 2007

Equals inverse binomial transform of A000108 starting (1, 2, 5, 14, 42,...). - Gary W. Adamson, Dec 10 2007

Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example, Phi([1]) is the Catalan numbers A000108. The present sequence is Phi([0,1,1]), see the 6th formula. - Gary W. Adamson, Oct 27 2008

G.f.: 1/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/.... (continued fraction). - Paul Barry, Dec 06 2008

G.f.: 1/(1-(x+x^2)/(1-x^2/(1-(x+x^2)/(1-x^2/(1-(x+x^2)/(1-x^2/(1-.... (continued fraction). - Paul Barry, Feb 08 2009

a(n) = (-3)^(1/2)/(6*(n+2)) * (-1)^n*(3*hypergeom([1/2, n+1],[1],4/3) - hypergeom([1/2, n+2],[1],4/3)). - Mark van Hoeij, Nov 12 2009

G.f.: 1/(1-x/(1-x/(1-x^2/(1-x/(1-x/(1-x^2/(1-x/(1-x/(1-x^2/(1-... (continued fraction). - Paul Barry, Mar 02 2010

G.f.: 1/(1-x/(1-x/(1+x-x/(1-x/(1+x-x/(1-x/(1+x-x/(1-x/(1+x-x/(1-... (continued fraction). - Paul Barry, Jan 26 2011 [Adds apparently a third '1' in front. - R. J. Mathar, Jan 29 2011]

Let A(x) be the g.f., then B(x)=1+x*A(x) = 1 +1*x +1*x^2 +2*x^3 +4*x^4 +9*x^5 +... = 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1+x) (continued fraction); more generally B(x)=C(x/(1+x)) where C(x) is the g.f. for the Catalan numbers (A000108). - Joerg Arndt, Mar 18 2011

a(n) = (2/Pi)*integral(x=-1..1, (1+2*x)^n*sqrt(1-x^2)). - Peter Luschny, Sep 11 2011

G.f.: (1-x-sqrt(1-2*x-3*(x^2)))/(2*(x^2)) = 1/2/(x^2)-1/2/x-1/2/(x^2)*G(0); G(k) = 1+(4*k-1)*x*(2+3*x)/(4*k+2-x*(2+3*x)*(4*k+1)*(4*k+2) /(x*(2+3*x)*(4*k+1)+(4*k+4)/G(k+1)), if -1 < x < 1/3; (continued fraction). - Sergei N. Gladkovskii, Dec 01 2011

G.f.: (1-x-sqrt(1-2*x-3*(x^2)))/(2*(x^2)) = (-1 + 1/G(0))/(2*x) ; G(k) = 1-2*x/(1+x/(1+x/(1-2*x/(1-x/(2-x/G(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011

0 = a(n) * (9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1) * ( -3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2) * (a(n+2) + a(n+3)) unless n=-2. - Michael Somos, Mar 23 2012

a(n) = (-1)^n*hypergeometric([-n,3/2],[3],4). - Peter Luschny, Aug 15 2012

Representation in terms of special values of Jacobi polynomials P(n,alpha,beta,x), in Maple notation: a(n)= 2*(-1)^n*n!*JacobiP(n,2,-3/2-n,-7)/(n+2)!, n>=0. - Karol A. Penson, Jun 24 2013

G.f.: Q(0)/x - 1/x, where Q(k) = 1 + (4*k+1)*x/((1+x)*(k+1) - x*(1+x)*(2*k+2)*(4*k+3)/(x*(8*k+6)+(2*k+3)*(1+x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013

Catalan(n+1) = Sum_{k=0..n} binomial(n,k)*a(k). E.g.: 42 = 1*1 + 4*1 + 6*2 + 4*4 + 1*9. Doron Zeilberger, Mar 12, 2015.

G.f. A(x) with offset 1 satisfies: A(x)^2 = A( x^2/(1-2*x) ). - Paul D. Hanna, Nov 08 2015

Conjecture: +(n+2)*a(n) +(-2*n-1)*a(n-1) +3*(-n+1)*a(n-2)=0. - R. J. Mathar, Sep 06 2016

a(n) = GegenbauerPoly(n,-n-1,-1/2)/(n+1). - Emanuele Munarini, Oct 20 2016


G.f.: 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 51*x^6 + 127*x^7 + 323*x^8 + ...


Three different Maple scripts for this sequence:

[seq(add(binomial(n+1, k)*binomial(n+1-k, k-1), k=0..ceil((n+1)/2))/(n+1), n=0..50)];

A001006 := proc(n) option remember; local k; if n <= 1 then 1 else procname(n-1) + add(procname(k)*procname(n-k-2), k=0..n-2); fi; end;

Order := 20: solve(series(x/(1+x+x^2), x)=y, x);

zl:=4*(1-z+sqrt(1-2*z-3*z^2))/(1-z+sqrt(1-2*z-3*z^2))^2/2: gser:=series(zl, z=0, 35): seq(coeff(gser, z, n), n=0..29); # Zerinvary Lajos, Feb 28 2007

# n -> [a(0), a(1), .., a(n)]

A001006_list := proc(n) local w, m, j, i; w := proc(i, j, n) option remember;

if min(i, j, n) < 0 or max(i, j) > n then 0

elif n = 0 then if i = 0 and j = 0 then 1 else 0 fi else

w(i, j + 1, n - 1) + w(i - 1, j, n - 1) + w(i + 1, j - 1, n - 1) fi end:

[seq( add( add( w(i, j, m), i = 0..m), j = 0..m), m = 0..n)] end:

A001006_list(29); # Peter Luschny, May 21 2011


a[0] = 1; a[n_Integer] := a[n] = a[n - 1] + Sum[a[k] * a[n - 2 - k], {k, 0, n - 2}]; Array[a[#] &, 30]

CoefficientList[Series[(1 - x - (1 - 2x - 3x^2)^(1/2))/(2x^2), {x, 0, 29}], x] (* Jean-François Alcover, Nov 29 2011 *)

Table[Hypergeometric2F1[(1-n)/2, -n/2, 2, 4], {n, 0, 29}] (* Peter Luschny, May 15 2016 *)

Table[GegenbauerC[n, -n-1, -1/2]/(n+1), {n, 0, 100}] (* Emanuele Munarini, Oct 20 2016 *)


(PARI) {a(n) = polcoeff( ( 1 - x - sqrt((1 - x)^2 - 4 * x^2 + x^3 * O(x^n))) / (2 * x^2), n)}; /* Michael Somos, Sep 25 2003 */

(PARI) {a(n) = if( n<0, 0, n++; polcoeff( serreverse( x / (1 + x + x^2) + x * O(x^n)), n))}; /* Michael Somos, Sep 25 2003 */

(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp(x + x * O(x^n)) * besseli(1, 2 * x + x * O(x^n)), n))}; /* Michael Somos, Sep 25 2003 */

(Maxima) a[0]:1$



makelist(a[n], n, 0, 12); /* Emanuele Munarini, Mar 02 2011 */


M(n) := coeff(expand((1+x+x^2)^(n+1)), x^n)/(n+1);

makelist(M(n), n, 0, 60); /* Emanuele Munarini, Apr 04 2012 */

(Maxima) makelist(ultraspherical(n, -n-1, -1/2)/(n+1), n, 0, 12); /* Emanuele Munarini, Oct 20 2016 */


a001006 n = a001006_list !! n

a001006_list = zipWith (+) a005043_list $ tail a005043_list

-- Reinhard Zumkeller, Jan 31 2012


from gmpy2 import divexact

A001006 = [1, 1]

for n in range(2, 10**3):

....A001006.append(divexact(A001006[-1]*(2*n+1)+(3*n-3)*A001006[-2], n+2))

# Chai Wah Wu, Sep 01 2014


def mot():

    a, b, n = 0, 1, 1

    while True:

        yield b//n

        n += 1

        a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1))

A001006 = mot()

print([A001006.next() for n in range(30)]) # Peter Luschny, May 16 2016


Cf. A026300, A005717, A020474, A001850, A004148. First column of A064191, A064189, A000108, A088615, A007971, A001405, A005817, A049401, A007579, A007578, A097862, A144218, A005773, A178515, A217275. First row of A064645.

Bisections: A026945, A099250.

Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

a(n) = A005043(n)+A005043(n+1).

A086246 is another version, although this is the main entry. Column k=3 of A182172.

When read mod 2,3,5,7,11: A039963, A039964, A258712, A258711, A258710.

Cf. A004148, A004149, A023421, A023422, A023423.

Sequence in context: A168051 A166587 A168049 * A086246 A247100 A230556

Adjacent sequences:  A001003 A001004 A001005 * A001007 A001008 A001009




N. J. A. Sloane



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Last modified May 27 07:59 EDT 2017. Contains 287203 sequences.