A001006 -id:A001006 - cp's OEIS frontend

A026945 A bisection of the Motzkin numbers A001006.

1, 2, 9, 51, 323, 2188, 15511, 113634, 853467, 6536382, 50852019, 400763223, 3192727797, 25669818476, 208023278209, 1697385471211, 13933569346707, 114988706524270, 953467954114363, 7939655757745265, 66368199913921497

Offset: 0

Clark Kimberling

a(n) is the sum of the squares of numbers in row n of array T given by A026300.
Number of closed walks of length 2n on the one-way infinite ladder graph starting from (and ending at) a node of degree 2. - Mitch Harris, Mar 06 2004
a(n) is the number of ways to connect 2n points labeled 1,2,...,2n in a line with 0 or more noncrossing arcs. For example, with arcs separated by dashes, a(2)=9 counts {} (no arcs), 12, 13, 14, 23, 24, 34, 12-34, 14-23. - David Callan, Sep 18 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Igor Dolinka, James East, Athanasios Evangelou, Desmond FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin, and James Mitchell, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv preprint arXiv:1507.04838 [math.CO], 2015-2018.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
Rosa Orellana, Nancy Wallace, and Mike Zabrocki, Quasipartition and planar quasipartition algebras, Sém. Lotharingien Comb., Proc. 36th Conf. Formal Power Series Alg. Comb. (2024) Vol. 91B, Art. No. 50. See p. 7.
Willow Stewart and Daniel Tubbenhauer, Representation gaps of rigid planar diagram monoids, arXiv:2505.05846 [math.RT], 2025. See references.
Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).

Cf. A001006, A099250.

G:=(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2): GG:=series(G,x=0,60): 1, seq(coeff(GG,x^(2*n)),n=1..23);
a := n -> hypergeom([1/2-n, -n], [2], 4);
seq(simplify(a(n)), n=0..29); # Peter Luschny, May 15 2016

Table[SeriesCoefficient[(1-x-Sqrt[1-2*x-3*x^2])/(2*x^2),{x,0,2*n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)
MotzkinNumber = DifferenceRoot[Function[{y, n}, {(-3n-3)*y[n] + (-2n-5)*y[n+1] + (n+4)*y[n+2] == 0, y[0] == 1, y[1] == 1}]];
Table[MotzkinNumber[2n], {n, 0, 20}] (* Jean-François Alcover, Oct 27 2021 *)

C(n)=binomial(2*n,n)/(n+1);
a(n)=sum(k=0,n, binomial(2*n,2*k)*C(k));
\\ Joerg Arndt, May 04 2013

{a(n)=polcoeff(1/x*serreverse( x * (1-x) * (1-2*x)^2 /(1 - 3*x + 3*x^2 +x^2*O(x^n)) ),n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Oct 03 2014

a(n) = A005043(2n) + A005043(2n+1). - Ralf Stephan, Feb 06 2004
a(n) = Sum_{k=0..n} binomial(2n,2k)*C(k), C(n)=A000108(n); - Paul Barry, Jul 11 2008
a(n) = (2/Pi)*integral(x=-1..1, (1+2*x)^(2*n)*sqrt(1-x^2)). - Peter Luschny, Sep 11 2011
D-finite with recurrence: (n+1)*(2*n+1)*a(n) = (14*n^2+9*n-2)*a(n-1) + 3*(14*n^2-51*n+43)*a(n-2) - 27*(n-2)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ 3^(2*n+3/2)/(2^(5/2)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012
G.f.: (1/x) * Series_Reversion( x * (1-x) * (1-2*x)^2 / (1 - 3*x + 3*x^2) ). - Paul D. Hanna, Oct 03 2014
From Peter Luschny, May 15 2016: (Start)
a(n) = ((9-9*n)*(2*n-3)*(4*n+1)*a(n-2)+((8*n-2))*(10*n^2-5*n-3)*a(n-1))/((1+2*n)*(4*n-3)*(n+1)) for n>=2.
a(n) =  hypergeom([1/2-n, -n], [2], 4). (End)

Entry revised by N. J. A. Sloane, Nov 16 2004

A099250 Bisection of Motzkin numbers A001006.

Original entry on oeis.org

1, 4, 21, 127, 835, 5798, 41835, 310572, 2356779, 18199284, 142547559, 1129760415, 9043402501, 73007772802, 593742784829, 4859761676391, 40002464776083, 330931069469828, 2750016719520991, 22944749046030949, 192137918101841817, 1614282136160911722

Offset: 0

N. J. A. Sloane, Nov 16 2004

a(n) is the number of grand Motzkin paths from (0,0) to (2n+2,0) that avoid vertices (k,0) for all odd k and end on a down step. - Alexander Burstein, May 11 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
Veronika Irvine, Stephen Melczer, and Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018.

Cf. A001006, A026945, A048990.

G:=(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2): GG:=series(G,x=0,60): seq(coeff(GG,x^(2*n-1)),n=1..24);  # Emeric Deutsch
M := proc(n) option remember; `if`(n<2,1,(3*(n-1)*M(n-2)+(2*n+1)*M(n-1))/(n+2)) end: A099250 := n -> M(2*n+1):
seq(A099250(i),i=0..20); # Peter Luschny, Sep 11 2011

Take[CoefficientList[Series[(1-x-(1-2x-3x^2)^(1/2))/(2x^2), {x,0,60}], x], {2,-1,2}] (* Harvey P. Dale, Sep 11 2011 *)
Table[Hypergeometric2F1[-1/2-n, -n, 2, 4], {n, 0, 30}] (* Jean-François Alcover, Apr 03 2015 *)
MotzkinNumber = DifferenceRoot[Function[{y, n}, {(-3n-3)*y[n] + (-2n-5)*y[n+1] + (n+4)*y[n+2] == 0, y[0] == 1, y[1] == 1}]];
Table[MotzkinNumber[2n+1], {n, 0, 20}] (* Jean-François Alcover, Oct 27 2021 *)

my(x='x+O('x^66)); v=Vec((1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2)); vector(#v\2,n,v[2*n]) \\ Joerg Arndt, May 12 2013

{a(n)=polcoeff(1/x*serreverse( x*(1+x)/((1+2*x)^2*(1+x+x^2) +x^2*O(x^n)) ),n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Oct 03 2014

a(n) = (2/Pi)*Integral_{x=-1..1} (1+2*x)^(2*n+1)*sqrt(1-x^2). [Peter Luschny, Sep 11 2011]
Recurrence: (n+1)*(2*n+3)*a(n) = (14*n^2+23*n+6)*a(n-1) + 3*(14*n^2-37*n+21)*a(n-2) - 27*(n-2)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 3^(2*n+5/2)/(4*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
G.f.: (1/x) * Series_Reversion( x*(1+x) / ( (1+2*x)^2 * (1+x+x^2) ) ). - Paul D. Hanna, Oct 03 2014
From Peter Bala, Apr 20 2024: (Start)
a(n) = Sum_{k = 0..2*n+1} (-1)^(k+1) * binomial(2*n+1, k)*Catalan(k+1).
a(n) = Sum_{k = 0..2*n+1} (-1)^k * binomial(2*n+1, k)*Catalan(k+1)*3^(2*n-k+1).
(4*n - 1)*(2*n + 3)*(n + 1)*a(n) = 2*(4*n + 1)*(10*n^2 + 5*n - 3)*a(n-1) - 9*(4*n + 3)*(2*n - 1)*(n - 1)*a(n-2) with a(0) = 1 and a(1) = 4. (End)

More terms from Emeric Deutsch, Nov 17 2004

A026107 Second differences of Motzkin numbers (A001006).

Original entry on oeis.org

1, 3, 7, 18, 46, 120, 316, 841, 2257, 6103, 16611, 45475, 125139, 345957, 960417, 2676291, 7483299, 20989833, 59042805, 166520124, 470781528, 1333970190, 3787707322, 10775741271, 30711538351, 87677551081, 250704001213, 717923179762

Offset: 2

Clark Kimberling

nonn

Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1 = s(n), |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n-1), where T is array in A026105 and U(n,n+1), where U is array in A026120.
Also number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, s(n) = 0, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2.
Number of Motzkin paths of length n+1 that start with a (1,1) step and end with a (1,-1) step. - Emeric Deutsch, Jul 11 2001
Equals iterates of M * [1,1,1,1,0,0,0,...] where M = an infinite tridiagonal matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the super- and subdiagonals. - Gary W. Adamson, Jan 08 2009
Number of Motzkin paths of length n-1 that are allowed to go down to the line y=-1 [He-Shapiro, page 38]. - R. J. Mathar, Jul 23 2017
With offset 1, a[n] = [x^n](1 + x + x^2)^n - [x^(n-4)](1 + x + x^2)^n, that is, the difference between the n-th central trinomial coefficient and its fourth predecessor. For example, with n = 4, (1 + x + x^2)^4 = 1 + 4*x + 10*x^2 + 16*x^3 + 19*x^4 + 16*x^5 + 10*x^6 + 4*x^7 + x^8 and a(4) = 19 - 1. - David Callan, Dec 18 2021

T.-X. He and L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41.

Cf. A001006. First differences of A002026.

Cf. A026122.

a(n) = A001006(n+1) - 2*A001006(n) + A001006(n-1).
The sequence 1,1,3,7,18,... has a(n) = Sum_{k=0..n} binomial(n,2k)*A000108(k+1). - Paul Barry, Jul 18 2003
G.f.: ((1-z)^2*M - 1 + z - z^2)/z, where M is the generating function of the Motzkin sequence A001006 (M = 1 + z*M + z^2*M^2).
(n+3)*a(n) + 3*(-n-1)*a(n-1) + (-n-3)*a(n-2) + 3*(n-3)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012
a(n) ~ 2 * 3^(n + 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 17 2019
With offset 0 and a(0) = 1 prepended (see Paul Barry's formula above), a(n) = hypergeom([3/2, (1 - n)/2, -n/2], [1/2, 3], 4). - Peter Luschny, Dec 19 2021

Simpler definition from Ralf Stephan, Dec 16 2004

A120895 G.f. satisfies: A(x) = G(x)A(x^3G(x)^2) where G(x) is the g.f. of the Motzkin numbers (A001006).

Original entry on oeis.org

1, 1, 2, 5, 12, 30, 78, 206, 552, 1498, 4105, 11340, 31541, 88237, 248076, 700478, 1985397, 5646129, 16104378, 46056513, 132031176, 379315946, 1091890772, 3148736064, 9095091878, 26310816944, 76219704957, 221085782559, 642058752476, 1866693825362, 5432795508417

Offset: 0

Paul D. Hanna, Jul 14 2006

nonn

Equals column 0 and main diagonal of triangle A120894 (cascadence of 1+x+x^2).

			A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 30*x^5 + 78*x^6 + 206*x^7+...
= G(x)*A(x^3*G(x)^2) where
G(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 51*x^6 + 127*x^7 +...
is the g.f. of the Motzkin numbers (A001006) so that G(x) satisfies:
G(x) = 1 + x*G(x) + x^2*G(x)^2.

Cf. A120894, A120896, A120897; A001006; variants: A092684, A092687, A120899.

{a(n)=local(A=1+x,G=1/x*serreverse(x/(1+x+x^2+x*O(x^n)))); for(i=0,n,A=G*subst(A,x,x^3*G^2 +x*O(x^n)));polcoeff(A,n,x)}

A248137 Least positive integer m such that m + n divides M(m) + M(n), where M(.) is given by A001006.

Original entry on oeis.org

1, 1, 244, 1, 23, 4, 1, 1, 3494, 1, 68058, 4, 20, 18, 35, 1, 4, 14, 32, 13, 21, 1, 5, 22, 172, 7, 8, 1, 1, 28, 14, 19, 2, 178, 15, 227, 2, 6, 109, 1, 22, 122, 47, 22, 126, 1, 43, 60, 41, 18, 24, 1, 13, 23, 21, 24, 126, 1, 152, 6

Offset: 1

Zhi-Wei Sun, Oct 02 2014

nonn

Conjecture: a(n) exists for any n > 0.

			a(5) = 23 since 5 + 23 = 28 divides M(5) + M(23) = 21 + 1129760415 = 1129760436 = 28*40348587.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2187

Cf. A001006, A247824, A247937, A247940, A248124, A248125, A248133, A248136, A248139, A248142.

M[n_]:=Sum[Binomial[n,2k]Binomial[2k,k]/(k+1),{k,0,n/2}]
Do[m=1;Label[aa];If[Mod[M[m]+M[n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]

A167630 Riordan array (1/(1-x),xm(x)) where m(x) is the g.f. of Motzkin numbers A001006.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 8, 4, 1, 1, 17, 20, 13, 5, 1, 1, 38, 50, 38, 19, 6, 1, 1, 89, 126, 107, 63, 26, 7, 1, 1, 216, 322, 296, 196, 96, 34, 8, 1, 1, 539, 834, 814, 588, 326, 138, 43, 9, 1, 1, 1374, 2187, 2236, 1728, 1052, 507, 190, 53, 10, 1

Offset: 0

Philippe Deléham, Nov 07 2009

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  3,  1;
  1,  8,  8,  4,  1;
  1, 17, 20, 13,  5, 1;
  1, 38, 50, 38, 19, 6, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Antidiagonal sums give A082395.
Row sums give A383527.
Diagonals include: A006416, A034856, A086615, A140662.
Cf. A001006, A003462, A064189, A082397, A094531.

T:= proc(n, k) option remember; `if`(k=0, 1,
      `if`(k>n, 0, T(n-1, k-1)+T(n-1, k)+T(n-1, k+1)))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Apr 20 2018

T[, 0] = T[n, n_] = 1;
T[n_, k_] /; 0, ] = 0;
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 09 2019 *)

T(n,0)=1, T(0,k)=0 for k>0, T(n,k)=0 if k>n, T(n,k)=T(n-1,k-1)+T(n-1,k)+T(n-1,k+1).
Sum_{k=0..n} k * T(n,k) = A003462(n). - Alois P. Heinz, Apr 20 2018
Sum_{k=0..n} (-1)^(k+1) * T(n,k) = A082397(n-2) for n>=2. - Alois P. Heinz, May 02 2025

A116387 Expansion of 1/(sqrt(1-2x-3x^2)*(2-M(x))), where M(x) is the g.f. of the Motzkin numbers A001006.

Original entry on oeis.org

1, 2, 7, 22, 72, 234, 763, 2486, 8099, 26372, 85833, 279226, 907946, 2951066, 9587981, 31140034, 101104048, 328162170, 1064856217, 3454513274, 11204337056, 36332719182, 117795920249, 381848062066, 1237615088203, 4010710218384

Offset: 0

Paul Barry, Feb 12 2006

Binomial transform of A116383.

The substitution x-> x/(1+x+x^2) in the g.f. (this might be called an inverse Motzkin transform) yields the g.f. of A074331. - R. J. Mathar, Nov 10 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..200

List([0..30], n-> Sum([0..n], k-> Sum([0..n], j-> Binomial(n, j-k)*Binomial(j, n-j) ))); # G. C. Greubel, May 23 2019

[(&+[ (&+[Binomial(n, j-k)*Binomial(j, n-j): j in [0..n]]) : k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 23 2019

Table[Sum[Binomial[n,j-k]Binomial[j,n-j],{k,0,n},{j,0,n}],{n,0,30}] (* Harvey P. Dale, Feb 08 2012 *)

{a(n) = sum(k=0,n, sum(j=0,n, binomial(n, j-k)*binomial(j,n-j)))}; \\ G. C. Greubel, May 23 2019

[sum( sum(binomial(n, j-k)*binomial(j,n-j) for j in (0..n)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, May 23 2019

a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n,j-k)*C(j,n-j).
Conjecture: n*(17*n-142)*a(n) + (17*n^2 + 95*n + 138)*a(n-1) + (-391*n^2 + 2488*n - 2908)*a(n-2) + (-17*n^2 - 603*n + 1892)*a(n-3) + 2*(697*n-2021)*(n-4)*a(n-4) + 60*(17*n-47)*(n-4)*a(n-5) = 0. - R. J. Mathar, Nov 15 2011
a(n) ~ (1+sqrt(5))^n * (5+sqrt(5)) / 10. - Vaclav Kotesovec, Feb 08 2014

A152981 Sum of divisors of Motzkin number A001006(n).

Original entry on oeis.org

1, 1, 3, 7, 13, 32, 72, 128, 360, 1008, 3836, 9408, 15512, 66960, 252720, 785148, 1137960, 3340800, 13072776, 42465024, 69530400, 238761600, 678562560, 2412043920, 5270534880, 10943277120, 44922182340, 117229255200, 209990825568, 623486730240, 1698651339840, 7019676332760, 18578092462280

Offset: 0

Omar E. Pol, Dec 20 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 0..200

Cf. A000005, A000203, A001006, A152761.

A152981 := proc(n) numtheory[sigma](A001006(n)) ; end proc:
seq(A152981(n),n=0..40) ; # R. J. Mathar, Jul 08 2011

mot[0] = 1; mot[n_] := mot[n] = mot[n - 1] + Sum[mot[k] * mot[n - 2 - k], {k, 0, n - 2}]; Table[DivisorSigma[1, mot[n]], {n, 0, 32}] (* Amiram Eldar, Nov 26 2019 *)

a(n) = sigma(A001006(n)) = A000203(A001006(n)).

A152982 Sum of proper divisors of Motzkin number A001006(n).

Original entry on oeis.org

0, 0, 1, 3, 4, 11, 21, 1, 37, 173, 1648, 3610, 1, 25125, 139086, 474576, 284493, 984021, 6536394, 24265740, 18678381, 96214041, 277799337, 1282283505, 2077807083, 1899874619, 19252363864, 44221482398, 1967547359, 29743945411, 1265868629

Offset: 0

Omar E. Pol, Dec 20 2008

nonn

			a(6)=21 because A001006(6)=51, having as proper divisors 1, 3 and 17. - _Emeric Deutsch_, Dec 31 2008

Amiram Eldar, Table of n, a(n) for n = 0..200

Cf. A001065, A001006, A152762, A152981, A152983.

with(numtheory): M := proc (n) options operator, arrow: sum(binomial(n, 2*k)*binomial(2*k, k)/(k+1), k = 0 .. n) end proc: seq(sigma(M(n))-M(n), n = 0 .. 30); # Emeric Deutsch, Dec 31 2008

mot[0] = 1; mot[n_] := mot[n] = mot[n - 1] + Sum[mot[k] * mot[n - 2 - k], {k, 0, n - 2}]; propDivSum[n_] := DivisorSigma[1, n] - n; Table[propDivSum[mot[n]], {n, 0, 30}] (* Amiram Eldar, Nov 26 2019 *)

a(n) = A001065(A001006(n)).

Extended by Emeric Deutsch, Dec 31 2008

A152983 Number of divisors of Motzkin number A001006(n).

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 2, 4, 4, 6, 8, 2, 8, 24, 18, 4, 16, 8, 12, 16, 24, 48, 72, 12, 8, 6, 16, 8, 16, 8, 12, 4, 16, 64, 12, 2, 8, 8, 8, 8, 24, 96, 96, 6, 24, 72, 48, 24, 32, 128, 96, 16, 8, 8, 8, 16, 128, 60, 192, 6, 32, 32, 96, 8, 96, 512, 36, 24, 16, 24, 384, 24, 96, 144, 48, 64, 64, 32

Offset: 0

Omar E. Pol, Dec 20 2008

nonn

			a(5)=4 because the Motzkin number M(5)=21 has 4 divisors: 1,3,7 and 21. - _Emeric Deutsch_, Jan 14 2009

Amiram Eldar, Table of n, a(n) for n = 0..200

Cf. A000005, A001006, A152763.

with(numtheory): M := proc (n) options operator, arrow: (sum((-1)^j*binomial(n+1, j)*binomial(2*n-3*j, n), j = 0 .. floor((1/3)*n)))/(n+1) end proc: seq(tau(M(n)), n = 0 .. 82); # Emeric Deutsch, Jan 14 2009

mot[0] = 1; mot[n_] := mot[n] = mot[n - 1] + Sum[mot[k] * mot[n - 2 - k], {k, 0, n - 2}]; Table[DivisorSigma[0, mot[n]], {n, 0, 50}] (* Amiram Eldar, Nov 26 2019 *)

a(n) = A000005(A001006(n)).

Extended by Emeric Deutsch, Jan 14 2009

cp's OEIS Frontend

A026945 A bisection of the Motzkin numbers A001006.

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A099250 Bisection of Motzkin numbers A001006.

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A026107 Second differences of Motzkin numbers (A001006).

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A120895 G.f. satisfies: A(x) = G(x)*A(x^3*G(x)^2) where G(x) is the g.f. of the Motzkin numbers (A001006).

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A248137 Least positive integer m such that m + n divides M(m) + M(n), where M(.) is given by A001006.

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A167630 Riordan array (1/(1-x),xm(x)) where m(x) is the g.f. of Motzkin numbers A001006.

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A116387 Expansion of 1/(sqrt(1-2*x-3*x^2)*(2-M(x))), where M(x) is the g.f. of the Motzkin numbers A001006.

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A120895 G.f. satisfies: A(x) = G(x)A(x^3G(x)^2) where G(x) is the g.f. of the Motzkin numbers (A001006).

A116387 Expansion of 1/(sqrt(1-2x-3x^2)*(2-M(x))), where M(x) is the g.f. of the Motzkin numbers A001006.