0,5

I'm not sure "Motzkin sums" is a good name for this. It has the proprty that Motzkin(n) = A001006(n) = a(n) + a(n+1), e.g. A001006(4) = 9 = 3 + 6 = a(4) + a(5).

Number of 'Catalan partitions', that is partitions of a set 1,2,3,...,n into parts that are not singletons and whose convex hulls are disjoint when the points are arranged on a circle (so when the parts are all pairs we get Catalan numbers) - Aart Blokhuis (aartb(AT)win.tue.nl), Jul 04, 2000.

Number of ordered trees with n edges and no vertices of outdegree 1. For n > 1, number of dissections of a convex polygon by nonintersecting diagonals with a total number of n+1 edges. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 06 2002

Number of Motzkin paths of length n with no horizontal steps at level 0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 09 2003

Number of Dyck paths of semilength n with no peaks at odd level. Example: a(4)=3 because we have UUUUDDDD, UUDDUUDD and UUDUDUDD, where U=(1,1), D=(1,-1). Number of Dyck paths of semilength n with no ascents of length 1 (an ascent in a Dyck path is a maximal string of up steps). Example: a(4)=3 because we have UUUUDDDD, UUDDUUDD and UUDUUDDD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003

Arises in Schubert calculus as follows. Let P = complex projective space of dimension n+1. Take n projective subspaces of codimension 3 in P in general position. Then a(n) is the number of lines of P intersecting all these subspaces. - F. Hirzebruch, Feb 09, 2004

Difference between central trinomial coefficient and its predecessor. Example: a(6)=15=141-126 and (1+x+x^2)^6=...+ 126*x^5 + 141*x^6 +... (Catalan number A000108(n) is difference between central binomial coefficient and its predecessor.) - David Callan (callan(AT)stat.wisc.edu), Feb 07 2004

First diagonal of triangular array in A059346.

a(n) = number of 321-avoiding permutations on [n] in which each left-to-right maximum is a descent (i.e. is followed by a smaller number). For example, a(4) counts 4123, 3142, 2143. - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005

The Hankel transform of this sequence give A000012 = [1, 1, 1, 1, 1, 1, 1, . . .]; example : Det([1, 0, 1, 1; 0, 1, 1, 3; 1, 1, 3, 6; 1, 3, 6, 15]) = 1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 28 2005

The number of projective invariants of degree 2 for n labeled points on the projective line. - Benjamin J. Howard (bhoward(AT)ima.umn.edu), Nov 24 2006

Define a random variable X=trA^2, where A is a 2 X 2 unitary symplectic matrix chosen from USp(2) with Haar measure. The n-th central moment of X is E[(X+1)^n] = a(n). - Andrew V. Sutherland (drew(AT)math.mit.edu), Dec 02 2007

Contribution from Samson Black (sblack1(AT)uoregon.edu), Aug 27 2008: (Start)

Let V be the adjoint representation of the complex Lie algebra sl(2). The dimension of the invariant subspace of the n-th tensor power of V is a(n).

(End)

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 08 2009: (Start)

Starting with offset 3 = iterates of M * [1,1,1,...], where M = a tridiagonal

matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. (End)

Binomial transform of A126930. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 26 2009]

M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563.

Almkvist, Gert; Dicks, Warren; Formanek, Edward; Hilbert series of fixed free algebras and noncommutative classical invariant theory. J. Algebra 93 (1985), no. 1, 189-214.

D. L. Andrews and T. Thirunamachandran, On three-dimensional rotational averages, J. Chem. Phys., 67 (1977), 5026-5033.

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.

R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.

P. Hanlon, Counting interval graphs. Trans. Amer. Math. Soc. 272 (1982), no. 2, 383-426.

B. Howard, J. Millson, A. Snowden and R. Vakil, http://arXiv.org/abs/math.AG/0505096, "The projective invariants of ordered points on the line", preprint, 2005.

D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

J. Riordan, Enumeration of plane trees by branches and endpoints, J. Combin. Theory, A 23 (1975), 214-222.

E. Royer, Interpretation combinatoire des moments negatifs des valeurs de fonctions L au bord de la bande critique, Ann. Sci. Ecole Norm. Sup. (4) 36 (2003), no. 4, 601-620.

H. C. H. Schubert, Math. Annalen, 1895.

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

D.-N. Verma (dnverma(AT)math.tifr.res.in), Towards Classifying Finite Point-Set Configurations, preprint, 1997.

F. Yano and H. Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.

T. D. Noe, Table of n, a(n) for n=0..200

David Callan, Riordan numbers are differences of trinomial coefficients.

W. Y. C. Chen, E. Y. P. Deng and L. L. M. Yang, Riordan paths and derangements

E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 423

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, 2008.

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

E. Royer, Interpretation combinatoire des moments negatifs des valeurs de fonctions L au bord de la bande critique

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

a(n)=sum((-1)^(n-k)*binomial(n, k)*A000108(k), k=0..n). a(n)=sum(binomial(n+1, k)*binomial(n-k-1, k-1), k=0..ceil(n/2))/(n+1); (for n>1) - Len Smiley (smiley(AT)math.uaa.alaska.edu)

G.f.: (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)).

G.f.: 2/(1+x+sqrt(1-2*x-3*x^2)) - Paul Peart (ppeart(AT)fac.howard.edu), May 27, 2000

a(n+1) + (-1)^n = a(0)*a(n) + a(1)*a(n-1) + ... + a(n)*a(0) - Bernhart.

a(n) = (1/(n+1)) * Sum_{i} (-1)^i*binomial(n+1, i)*binomial(2*n-2*i, n-i) - Bernhart.

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

E.g.f.: exp(x)*(BesselI(0, 2*x)-BesselI(1, 2*x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 28 2003

a(n)=A001006(n-1)-a(n-1).

a(n+1) = Sum[k=0..n, (-1)^k*A026300(n, k) ], where A026300 is the Motzkin triangle.

a(n)=sum{k=0..n, (-1)^k*binomial(n, k)*binomial(k, floor(k/2))} - Paul Barry (pbarry(AT)wit.ie), Jan 27 2005

a(n) = Sum_{k>=0} A086810(n-k, k) . - Philippe DELEHAM, May 30 2005

a(n+2) = Sum_{ k>=0} A064189(n-k, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 31 2005

Moment representation: a(n)=(1/(2*pi))*Int(x^n*sqrt((1+x)(3-x))/(1+x),x,-1,3); - Paul Barry (pbarry(AT)wit.ie), Jul 09 2006

Inverse binomial transform of A000108 (Catalan numbers) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 20 2006

a(n) = (2/pi)* Integral_{t_0..pi} (4cos^2(x)-1)^n*sin^2(x)dx - Andrew V. Sutherland (drew(AT)math.mit.edu), Dec 02 2007

G.f.: 1/(1-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-.....(continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Jan 22 2009]

G.f.: 1/(1+x-x/(1-x/(1+x-x/(1-x/(1+x-x/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), May 16 2009]

a(5)=6 because the only dissections of a polygon with a total number of 6 edges are: five pentagons with one of the five diagonals and the hexagon with no diagonals.

A005043 := proc(n) option remember; if n <= 1 then 1-n else (n-1)*(2*A005043(n-1)+3*A005043(n-2))/(n+1); fi; end;

Order := 20: solve(series((x-x^2)/(1-x+x^2), x)=y, x); #outputs g.f.

a[0] = 1; a[1] = 0; a[n_] := a[n] = (n - 1)*(2*a[n - 1] + 3*a[n - 2])/(n + 1); Table[ a[n], {n, 0, 30}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 14 2005)

(PARI) a(n)=if(n<0, 0, n++; polcoeff(serreverse((x-x^3)/(1+x^3)+x*O(x^n)), n)) /* Michael Somos May 31 2005 */

Row sums of triangle A020474, first differences of A082395.

Cf. A126120.

Sequence in context: A017924 A052827 A033192 this_sequence A099323 A058534 A063778

Adjacent sequences: A005040 A005041 A005042 this_sequence A005044 A005045 A005046

nonn,easy,nice,new

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

Thanks to Laura L. M. Yang (yanglm(AT)hotmail.com) for a correction, Aug 29, 2004