A126120 -id:A126120 - cp's OEIS frontend

A321396 Square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 0, related to a class of Motzkin trees.

0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 1, 3, 2, 5, 0, 1, 0, 1, 1, 3, 2, 7, 0, 0, 1, 0, 1, 1, 3, 3, 9, 5, 14, 0, 1, 0, 1, 1, 3, 3, 9, 7, 20, 0, 0, 1, 0, 1, 1, 3, 3, 10, 9, 27, 19, 42

Offset: 0

Peter Luschny, Nov 11 2018

The recursively specified combinatorial structure related to the array is the set of Motzkin trees where all leaves are at the same unary height (see section 3.2 in O. Bodini et al.).

			Array begins:
    [0]  0, 1, 0, 1, 0, 2, 0,  5,  0, 14,  0,  42,   0, 132, ...  A126120
    [1]  0, 1, 0, 1, 1, 2, 2,  7,  5, 20, 19,  60,  62, 202, ...  A300126
    [2]  0, 1, 0, 1, 1, 3, 2,  9,  7, 27, 25,  85,  86, 287, ...  A321572
    [3]  0, 1, 0, 1, 1, 3, 3,  9,  9, 29, 32,  93, 111, 317, ...
    [4]  0, 1, 0, 1, 1, 3, 3, 10,  9, 31, 34, 100, 119, 344, ...
    [5]  0, 1, 0, 1, 1, 3, 3, 10, 10, 31, 36, 102, 126, 352, ...
    [6]  0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 36, 104, 128, 359, ...
    [7]  0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 37, 104, 130, 361, ...
    [8]  0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 37, 105, 130, 363, ...
    [9]  0, 1, 0, 1, 1, 3, 3, 10, 10, 32, 37, 105, 131, 363, ...
Array read by ascending diagonals:
    [0]  0
    [1]  0, 1
    [2]  0, 1, 0
    [3]  0, 1, 0, 1
    [4]  0, 1, 0, 1, 0
    [5]  0, 1, 0, 1, 1, 2
    [6]  0, 1, 0, 1, 1, 2, 0
    [7]  0, 1, 0, 1, 1, 3, 2, 5
    [8]  0, 1, 0, 1, 1, 3, 2, 7, 0
    [9]  0, 1, 0, 1, 1, 3, 3, 9, 5, 14

Olivier Bodini, Danièle Gardy, Bernhard Gittenberger, Zbigniew Gołębiewski, On the number of unary-binary tree-like structures with restrictions on the unary height, arXiv:1510.01167v1 [math.CO], 2015.

Cf. A321395 (antidiagonal sums), A321397 (limit).

Cf. A000108 (Catalan), A001006 (Motzkin), A126120 (binary Catalan trees, row 0), A300126 (row 1), A321572 (row 2).

Arow := proc(n, len) local rowgf, ser;
rowgf := proc(n) option remember; `if`(n = 0, (1-sqrt(1-4*z^2))/(2*z),
expand((1 - sqrt(1 - 4*z^2*rowgf(n-1)))/(2*z))) end:
ser := series(rowgf(n)/z^n, z, 2*(2+max(len, n)));
seq(coeff(ser, z, k), k=0..len) end:
seq(Arow(n, 13), n=0..9);

nmax = 11; gf[-1] = 1; gf[n_] := gf[n] = (1-Sqrt[1 - 4z^2 gf[n-1]])/(2z);
row[n_] := row[n] = gf[n]/z^n + O[z]^(nmax+1) // CoefficientList[#, z]&;
A[n_, k_] := row[n][[k + 1]];
Table[A[n - k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 08 2018 *)

Define a sequence of generating functions recursively gf(-1) = 1 and for n >= 0
gf(n) = (1 - sqrt(1 - 4*z^2*gf(n-1)))/(2*z).
Row n of the array has the generating function gf(n)/z^n. For fixed k column k differs only for finitely many indices from the limit value A321397(k).

A121448 Triangle read by rows: T(n,k) is the number of binary trees with n edges and having k vertices of outdegree 1 (n>=0, k>=0). A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.

Original entry on oeis.org

1, 0, 2, 1, 0, 4, 0, 6, 0, 8, 2, 0, 24, 0, 16, 0, 20, 0, 80, 0, 32, 5, 0, 120, 0, 240, 0, 64, 0, 70, 0, 560, 0, 672, 0, 128, 14, 0, 560, 0, 2240, 0, 1792, 0, 256, 0, 252, 0, 3360, 0, 8064, 0, 4608, 0, 512, 42, 0, 2520, 0, 16800, 0, 26880, 0, 11520, 0, 1024, 0, 924, 0, 18480, 0

Offset: 0

Emeric Deutsch, Jul 31 2006

T(2n,0) = binomial(2n,n)/(n+1) (the Catalan numbers; A000108); T(2n+1,0)=0. T(n,n)=2^n (A000079). Sum(k*T(n,k),k=0..n)=2*binomial(2n,n-1)=2*A001791(n). After deleting the zeros, reflection of A091894.
From Tom Copeland, Feb 07 2016: (Start)
A shifted o.g.f. is OG(x,t) = [1 - 2tx - sqrt[(1-2tx)^2-4x^2]] / (2x) = x + 2t x^2 + (1+4t^2) x^3 + ... with compositional inverse OGinv(x,t) =  x / (1 + 2tx + x^2), the shifted o.g.f. for A053117 (mod signs).
For x > 0 and choosing the positive square root, OG(x^2,t) = H(x,t) = x^2 + 2t x^4 + (1+4t^2) x^6 + ... has the compositional inverse Hinv(x,t) = sqrt[x / (1 + 2tx + x^2)] , which satisfies Hinv(H(x, t), t) = x, and which is the generating function for the Legendre polynomials (mod signs, cf. A008316) times sqrt(x).
In general, GB(x,t,b) =  [x / (1 - 2tx + x^2)]^b is a generator for the Gegenbauer polynomials times x^b for positive roots with compositional inverse about the origin GBinv(x,t,b) = OG(x^(1/b),-t) for x>0. Cf. A097610.
(End)
From Tom Copeland, Feb 09 2016: (Start)
z1 = OG(x,t) is the zero that vanishes for x=0 for the quadratic polynomial Q(z;z1(x,t),z2(x,t)) =(z-z1)(z-z2) = z^2 - (z1+z2) z + (z1*z2) = z^2 - e1 z + e2 = z^2 - [(1-2tx)/x] z + 1, where e1 and e2 are the elementary symmetric polynomials for two indeterminates.
The other zero is given by z2(x,t) = [1 - 2tx + sqrt[(1-2tx)^2-4x^2]] / (2x) = (1 - 2tx)/x - z1(x,t).
The two are zeros of the elliptic curve in Legendre normal form y^2 = z (z-z1)(z-z2). (Added Feb 13 2016. See Landweber et al., p 14. Cf. A097610.)
(End)

			T(2,2)=4 because, denoting by L (R) an edge going from a vertex to a left (right) child, we have the paths: LL, LR, RL and RR.
Triangle starts:
  1;
  0,2;
  1,0,4;
  0,6,0,8;
  2,0,24,0,16;

Colin Defant, Postorder Preimages, arXiv preprint arXiv:1604.01723 [math.CO], 2016.
FindStat - Combinatorial Statistic Finder, The number of vertices with out-degree 1 in a binary tree.
P. Landweber, D. Ravenel, and R. Stong, Periodic cohomology theories defined by elliptic curves

Cf. A000108, A000079, A001791, A091894.
Cf. A038207, A053117, A097610, A126120, A145271, A180874, A218272.
Cf. A008316.

T:=proc(n,k) if n-k mod 2 = 0 then 2^k*binomial(n+1,k)*binomial(n+1-k,(n-k)/2)/(n+1) else 0 fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

nn=10;Drop[CoefficientList[Series[(1-2x y - ((-4x^2+(1-2x y)^2))^(1/2))/(2 x),{x,0,nn}],{x,y}],1]//Grid  (* Geoffrey Critzer, Feb 20 2013 *)

T(n,k) = 2^k*binomial(n+1,k)binomial(n+1-k,(n-k)/2)/(n+1) if n-k is even; otherwise, T(n,k) = 0. G.f. G=G(t,z) satisfies G=1+2tzG+z^2*G^2.
T(n,k) = 2^k*A097610(n,k). - Philippe Deléham, Aug 17 2006
From Tom Copeland, Feb 09 2016: (Start)
The following is from the formalism in A097610 with h1 = 2t, h2 = 1, and MT(n,h1,h2) = MT(n,2t,1) and with OG(x,t) defined above.
E.g.f.: M(x,t) = e^(2tx) AC(x) = exp[x MT(.,2t,1)] = exp[x P(.,t)], where AC(x) = I_1(2x)/x = Sum_{n>=0} x^(2n)/(n!(n+1)!) = exp(c.x) is the e.g.f. of A126120.
P(n,t) = MT(n,2t,1) = (c. + 2t)^n = Sum_{k=0..n} binomial(n,k) c(n-k) (2t)^k with c(k) = A126120(k). P(n,t+s) = (c. + 2t + 2s)^n = (P(.,t) + 2s)^n.
P(n,t) = t^n FC(n,c./t) = t^n (2 + c./t)^n, where FC(n,t) = (2 + t)^n are the face polynomials (vectors) of the hypercubes of A038207, i.e., the row polynomials of this entry can be obtained as the umbral composition of the reverse face polynomials with the aerated Catalan numbers of A000108.
The lowering and raising operators for the row polynomials P(n,t) of this entry are L = (1/2) d/dt = (1/2) D and R = 2t + dlog{AC(L)}/dL = 2t + Sum_{n>=0} b(n) L^(2n+1)/(2n+1)! = 2t + L - L^3/3! + 5 L^5/5! - ...  with b(n) = (-1)^n A180874(n+1).
Let CP(n,t) = P(n+1,t) with CP(0,t) = 0. Then the infinitesimal generator for CP(n,t) is g(x) d/dx with g(x) = 1 /[dOGinv(x,t)/dx] = x^2 / [(OGinv(x,t))^2 (1 - x^2)] = (1 + 2t x + x^2)^2 / (1 - x^2) so that [g(x)d/dx]^n/n! x evaluated at x = 0 gives the row polynomial CP(n,t), i.e., exp[x g(u)d/du] u |_(u=0) = OG(x,t) = 1 /[1 - x P(.,t)]. Cf. A145271.
g(x) = 1 + 4t x + (3+4t) x^2 + 8t x^3 + 4(1+t^2) x^4 + 8t x^5 + 4(1+t^2) x^6 + 8t x^7 + ... has the repeating coefficients of the vector V = (1, 4t, 3+4t, 8t, 4(1+t^2), 8t, 4(1+t^2), 8t, ...). Form the lower triangular matrix U with all ones on the diagonal and below. Multiply the n-th diagonal of U by V(n), giving the matrix VU with VU(n,k) = V(n-k). Then (1,0,0,0,..) [VU * DM]^n/n! (0,1,0,0,..)^T = CP(n,t) = P(n-1,t) for n>0 with DM being the matrix A218272 representing differentiation of a power series.
(End)

A138350 Moment sequence of tr(A^2) in USp(4).

Original entry on oeis.org

1, -1, 3, -6, 20, -50, 175, -490, 1764, -5292, 19404, -60984, 226512, -736164, 2760615, -9202050, 34763300, -118195220, 449141836, -1551580888, 5924217936, -20734762776, 79483257308, -281248448936, 1081724803600, -3863302870000, 14901311070000, -53644719852000

Offset: 0

Andrew V. Sutherland, Mar 16 2008

sign

If A is a random matrix in the compact group USp(4) (4 X 4 complex matrices which are unitary and symplectic), then a(n) = E[(tr(A^2))^n] is the n-th moment of the trace of A^2. See A138351 for central moments.

			a(5) = -50 because E[(tr(A^2))^5] = -50 for a random matrix A in USp(4).
a(5) = A126120(5)*A138364(6)-A138364(5)*A126120(6) = 0*0-10*5 = -50.

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, Massachusetts Institute of Technology. Dept. of Mathematics.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT]

A signed version of A005558, which is the main entry for this sequence.

Cf. A138349, A138351.

a[n_] := 1/2*Binomial[2*Floor[n/2]+1, Floor[n/2]+1]*CatalanNumber[1/2*(n+Mod[n, 2])]*(Mod[n, 2]+2); Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Mar 13 2014 *)

a(n)=(1/2)Integral_{x=0..Pi,y=0..Pi}(2cos(2x)+2cos(2y))^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy. a(n)=A126120(n)A138364(n+1)-A138364(n)A126120(n+1)

Conjectured e.g.f. BesselI[1,2x](BesselI[0,2x]-BesselI[1,2x])/x. - Benjamin Phillabaum, Feb 25 2011

A166135 Number of possible paths to each node that lies along the edge of a cut 4-nomial tree, that is rooted one unit from the cut.

Original entry on oeis.org

1, 1, 3, 7, 22, 65, 213, 693, 2352, 8034, 28014, 98505, 350548, 1256827, 4542395, 16517631, 60417708, 222087320, 820099720, 3040555978, 11314532376, 42243332130, 158196980682, 594075563613, 2236627194858, 8440468925400, 31921622746680, 120970706601255

Offset: 1

Rick Jarosh (rick(AT)jarosh.net), Oct 08 2009

nonn

This is the third member of an infinite series of infinite series, the first two being the Catalan and Motzkin integers. The Catalan numbers lie on the edge of cut 2-nomial trees, Motzkin integers on the edge of cut 3-nomial trees.
a(n) is the number of increasing unary-binary trees with associated permutation that avoids 213. For more information about increasing unary-binary trees with an associated permutation, see A245888. - Manda Riehl, Aug 07 2014
Number of positive walks with n steps {-2,-1,1,2} starting at the origin, ending at altitude 1, and staying strictly above the x-axis. - David Nguyen, Dec 16 2016

G. C. Greubel, Table of n, a(n) for n = 1..1000
Cyril Banderier, Christian Krattenthaler, Alan Krinik, Dmitry Kruchinin, Vladimir Kruchinin, David Tuan Nguyen, and Michael Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.
Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022).
Jérémie Bettinelli, Éric Fusy, Cécile Mailler, and Lucas Randazzo, A bijective study of Basketball walks, arXiv:1611.01478 [math.CO], 2016.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Rick Jarosh, Illustration of 4-nomial graph The series is the one at the top.
Rick Jarosh, First 4096 terms of the series in a text file.
Rick Jarosh, Illustrates the sequence in context. The above reference gives the first 16 terms of the first 128 sequences in the family, of which this sequence is the third, the first being the Catalan numbers, the second the Motzkin integers, the fourth A104632.

A055113 is the third sequence from the top of the graph illustrated above.

Cf. A000108, A001006, A069271, A092765, A126120.

[(&+[Binomial(n,k)*Binomial(n,2*n-3*k-1): k in [0..n]])/n : n in [1..30]]; // G. C. Greubel, Dec 12 2019

seq( add(binomial(n,k)*binomial(n,2*n-3*k-1), k=0..n)/n, n=1..30); # G. C. Greubel, Dec 12 2019

Rest[CoefficientList[Series[(Sqrt[(2-2Sqrt[1-4x]-3x)/x]-1)/2, {x, 0, 30}],x]] (* Benedict W. J. Irwin, Sep 24 2016 *)

vector(30, n, sum(k=0,n, binomial(n,k)*binomial(n,2*n-3*k-1))/n ) \\ G. C. Greubel, Dec 12 2019

[sum(binomial(n,k)*binomial(n,2*n-3*k-1) for k in (0..n))/n for n in (1..30)] # G. C. Greubel, Dec 12 2019

a(n) = ((36*n+18)*A092765(n) + (11*n+9)*A092765(n+1))/(2*(5*n+3)*(2*n+3)) (based on guessed recurrence). - Mark van Hoeij, Jul 14 2010
A(x) satisfies A(x)+A(x)^2 = A000108(x)-1, a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1) * C(2*n,n-k)*C(2*k-2,k-1). - Vladimir Kruchinin, May 12 2012
G.f.: (sqrt((2 - 2*sqrt(1-4*x) - 3*x)/x) - 1)/2. - Benedict W. J. Irwin, Sep 24 2016
a(n) ~ 4^n/(sqrt(5*Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 25 2016
Conjecture: 2*n*(2*n+1)*a(n) + (17*n^2-53*n+24)*a(n-1) + 6*(-13*n^2+43*n-36)*a(n-2) - 108*(2*n-5)*(n-3)*a(n-3) = 0. - R. J. Mathar, Oct 08 2016
a(n) = (1/n)*Sum_{k=0..n} binomial(n,k)*binomial(n,2*n-3*k-1). - David Nguyen, Dec 31 2016
From Alexander Burstein, Dec 12 2019: (Start)
1 + x*A(x) = 1/C(-x*C(x)^2), where C(x) is the g.f. of A000108.
F(x) = x*(1+x*A(x)) = x/C(-x*C(x)^2) is a pseudo-involution, i.e., the series reversion of x*(1 + x*A(x)) is x*(1 - x*A(-x)).
The B-sequence of F(x) is A069271, i.e., F(x) = x + x*F(x)*A069271(x*F(x)). (End)

A222006 Number of forests of rooted plane binary trees (all nodes have outdegree of 0 or 2) with n total nodes.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 10, 12, 27, 35, 79, 104, 244, 331, 789, 1083, 2615, 3652, 8880, 12523, 30657, 43661, 107326, 153985, 379945, 548776, 1357922, 1972153, 4892140, 7139850, 17747863, 26011843, 64776658, 95296413, 237689691, 350844814, 876313458, 1297367201, 3244521203, 4816399289

Offset: 0

Geoffrey Critzer, Feb 23 2013

nonn

Here, the binary trees are sized by total number of nodes.

			a(6) = 5: There is one forest with 6 trees, one forest with 4 trees and 3 forests with 2 trees, namely
1)...o..o..o..o..o..o...............
....................................
2)...o..o..o....o...................
.............../.\..................
..............o...o.................
....................................
3)...o........o.....................
..../.\....../.\....................
...o...o....o...o...................
....................................
4).....o....o.....5)......o.....o...
....../.\................/.\........
.....o...o..............o...o.......
..../.\..................../.\......
...o...o..................o...o.....

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Row sums of A342770.

b:= proc(n) option remember; `if`(irem(n, 2)=0, 0,
      `if`(n<2, n, add(b(i)*b(n-1-i), i=1..n-2)))
    end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(g(n-i*j, i-2)*binomial(b(i)+j-1, j), j=0..n/i)))
    end:
a:= n-> g(n, iquo(n-1, 2)*2+1):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 26 2013

nn=40;a=Drop[CoefficientList[Series[t=(1-(1-4x^2)^(1/2))/(2x),{x,0,nn}],x],1];CoefficientList[Series[Product[1/(1-x^i)^a[[i]],{i,1,nn-1}],{x,0,nn}],x]

O.g.f.: Product_{i>=1} 1/(1 - x^i)^A126120(i-1).

a(n) ~ c * 2^n / n^(3/2), where c = 1.165663931402962361339366557... if n is even, c = 1.490999501305559555120304528... if n is odd. - Vaclav Kotesovec, Aug 31 2014

A306437 Regular triangle read by rows where T(n,k) is the number of non-crossing set partitions of {1, ..., n} in which all blocks have size k.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 0, 0, 0, 1, 1, 5, 3, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 14, 0, 4, 0, 0, 0, 1, 1, 0, 12, 0, 0, 0, 0, 0, 1, 1, 42, 0, 0, 5, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 132, 55, 22, 0, 6, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 429, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Feb 15 2019

			Triangle begins:
  1
  1   1
  1   0   1
  1   2   0   1
  1   0   0   0   1
  1   5   3   0   0   1
  1   0   0   0   0   0   1
  1  14   0   4   0   0   0   1
  1   0  12   0   0   0   0   0   1
  1  42   0   0   5   0   0   0   0   1
  1   0   0   0   0   0   0   0   0   0   1
  1 132  55  22   0   6   0   0   0   0   0   1
Row 6 counts the following non-crossing set partitions (empty columns not shown):
  {{1}{2}{3}{4}{5}{6}}  {{12}{34}{56}}  {{123}{456}}  {{123456}}
                        {{12}{36}{45}}  {{126}{345}}
                        {{14}{23}{56}}  {{156}{234}}
                        {{16}{23}{45}}
                        {{16}{25}{34}}

Germain Kreweras, Sur les partitions non croisées d'un cycle, Discrete Math. 1 333-350 (1972).
Wikipedia, Noncrossing partition.

Row sums are A194560. Column k=2 is A126120. Trisection of column k=3 is A001764.

Cf. A000108, A000110, A000296, A001006, A001263, A001610, A016098, A038041, A061095, A125181, A134264.

T:= (n, k)-> `if`(irem(n, k)=0, binomial(n, n/k)/(n-n/k+1), 0):
seq(seq(T(n,k), k=1..n), n=1..14);  # Alois P. Heinz, Feb 16 2019

Table[Table[If[Divisible[n,d],d/n*Binomial[n,n/d-1],0],{d,n}],{n,15}]

If d|n, then T(n, d) = binomial(n, n/d)/(n - n/d + 1); otherwise T(n, k) = 0 [Theorem 1 of Kreweras].

A359364 Triangle read by rows. The Motzkin triangle, the coefficients of the Motzkin polynomials. M(n, k) = binomial(n, k) * CatalanNumber(k/2) if k is even, otherwise 0.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 0, 2, 1, 0, 10, 0, 10, 0, 1, 0, 15, 0, 30, 0, 5, 1, 0, 21, 0, 70, 0, 35, 0, 1, 0, 28, 0, 140, 0, 140, 0, 14, 1, 0, 36, 0, 252, 0, 420, 0, 126, 0, 1, 0, 45, 0, 420, 0, 1050, 0, 630, 0, 42, 1, 0, 55, 0, 660, 0, 2310, 0, 2310, 0, 462, 0

Offset: 0

Peter Luschny, Jan 09 2023

The generalized Motzkin numbers M(n, k) are a refinement of the Motzkin numbers M(n) (A001006) in the sense that they are coefficients of polynomials M(n, x) = Sum_{n..k} M(n, k) * x^k that take the value M(n) at x = 1. The coefficients of x^n are the aerated Catalan numbers A126120.
Variants are the irregular triangle A055151 with zeros deleted, A097610 with reversed rows, A107131 and A080159.
In the literature the name 'Motzkin triangle' is also used for the triangle A026300, which is generated from the powers of the generating function of the Motzkin numbers.

			Triangle M(n, k) starts:
[0] 1;
[1] 1, 0;
[2] 1, 0,  1;
[3] 1, 0,  3, 0;
[4] 1, 0,  6, 0,   2;
[5] 1, 0, 10, 0,  10, 0;
[6] 1, 0, 15, 0,  30, 0,   5;
[7] 1, 0, 21, 0,  70, 0,  35, 0;
[8] 1, 0, 28, 0, 140, 0, 140, 0,  14;
[9] 1, 0, 36, 0, 252, 0, 420, 0, 126, 0;

M. Artioli, G. Dattoli, S. Licciardi, and S. Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017, (Table 1 on p. 3).

Variants: A055151, A080159, A097610, A107131.
Columns M(n, 2*k): A000012, A000217, A034827, A000910, A088625, A088626, A213380.
Cf. A001006 (Motzkin numbers), A026300 (Motzkin gf. triangle), A126120 (aerated Catalan), A000108 (Catalan).
Cf. A138364, A107587, A002457, A002522, A025179, A025235, A056107, A014531, A023426, A091147, A189912, A343386, A343773, A359647, A359649.

CatalanNumber := n -> binomial(2*n, n)/(n + 1):
M := (n, k) -> ifelse(irem(k, 2) = 1, 0, CatalanNumber(k/2)*binomial(n, k)):
for n from 0 to 9 do seq(M(n, k), k = 0..n) od;
# Alternative, as coefficients of polynomials:
p := n -> hypergeom([(1 - n)/2, -n/2], [2], (2*x)^2):
seq(print(seq(coeff(simplify(p(n)), x, k), k = 0..n)), n = 0..9);
# Using the exponential generating function:
egf := exp(x)*BesselI(1, 2*x*t)/(x*t): ser:= series(egf, x, 11):
seq(print(seq(coeff(simplify(n!*coeff(ser, x, n)), t, k), k = 0..n)), n = 0..9);

from functools import cache
@cache
def M(n: int, k: int) -> int:
    if k %  2: return 0
    if n <  3: return 1
    if n == k: return (2 * (n - 1) * M(n - 2, n - 2)) // (n // 2 + 1)
    return (M(n - 1, k) * n) // (n - k)
for n in range(10): print([M(n, k) for k in range(n + 1)])

Let p(n, x) = hypergeom([(1 - n)/2, -n/2], [2], (2*x)^2).
p(n, 1) = A001006(n); p(n, sqrt(2)) = A025235(n); p(n, 2) = A091147(n).
p(2, n) = A002522(n); p(3, n) = A056107(n).
p(n, n) = A359649(n); 2^n*p(n, 1/2) = A000108(n+1).
M(n, k) = [x^k] p(n, x).
M(n, k) = [t^k] (n! * [x^n] exp(x) * BesselI(1, 2*t*x) / (t*x)).
M(n, k) = [t^k][x^n] ((1 - x - sqrt((x-1)^2 - (2*t*x)^2)) / (2*(t*x)^2)).
M(n, n) = A126120(n).
M(n, n-1) = A138364(n), the number of Motzkin n-paths with exactly one flat step.
M(2*n, 2*n) = A000108(n), the number of peakless Motzkin paths having a total of n up and level steps.
M(4*n, 2*n) = A359647(n), the central terms without zeros.
M(2*n+2, 2*n) = A002457(n) = (-4)^n * binomial(-3/2, n).
Sum_{k=0..n} M(n - k, k) = A023426(n).
Sum_{k=0..n} k * M(n, k) = 2*A014531(n-1) = 2*GegenbauerC(n - 2, -n, -1/2).
Sum_{k=0..n} i^k*M(n, k) = A343773(n), (i the imaginary unit), is the excess of the number of even Motzkin n-paths (A107587) over the odd ones (A343386).
Sum_{k=0..n} Sum_{j=0..k} M(n, j) = A189912(n).
Sum_{k=0..n} Sum_{j=0..k} M(n, n-j) = modified A025179(n).
For a recursion see the Python program.

A171368 Another version of A126216.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 5, 0, 9, 0, 1, 0, 0, 0, 21, 0, 14, 0, 1, 0, 0, 14, 0, 56, 0, 20, 0, 1, 0, 0, 0, 84, 0, 120, 0, 27, 0, 1, 0, 0, 42, 0, 300, 0, 225, 0, 35, 0, 1, 0, 0, 0, 330, 0, 825, 0, 385, 0, 44, 0, 1, 0, 0

Offset: 0

Philippe Deléham, Dec 06 2009

Expansion of the first column of the triangle T_(0,x), T_(x,y) defined in A039599; T_(0,0)= A053121, T_(0,1)= A089942, T_(0,2)= A126093, T_(0,3)= A126970.

T(n,k) is the number of Riordan paths of length n with k horizontal steps. A Riordan path is a Motzkin path with no horizontal steps on the x-axis. - Emanuele Munarini, Oct 14 2023

			Triangle begins:
  1 ;
  0,0 ;
  1,0,0 ;
  0,1,0,0 ;
  2,0,1,0,0 ;
  0,5,0,1,0,0 ;
  5,0,9,0,1,0,0 ;
  ...

Cf. A000108, A001003,

Sum_{k, 0<=k<=n} T(n,k)*x^k = A099323(n+1), A126120(n), A005043(n), A000957(n+1), A117641(n) for x = -1, 0, 1, 2, 3 respectively.

A202856 Moments of the quadratic coefficient of the characteristic polynomial of a random matrix in SU(2) X SU(2) (inside USp(4)).

Original entry on oeis.org

1, 2, 5, 14, 44, 152, 569, 2270, 9524, 41576, 187348, 866296, 4092400, 19684576, 96156649, 476038222, 2384463044, 12067926920, 61641751124, 317469893176, 1647261806128, 8605033903456, 45228349510660, 239061269168056, 1270130468349904, 6780349241182112, 36355025167014224, 195725149445320160, 1057729059593103808

Offset: 0

N. J. A. Sloane, Dec 25 2011

nonn

Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011-2012 (the sequence c-hat in Section 5.1.1).

Cf. A126869, A126120, A202814.

b:=n->coeff((x^2+1)^n, x, n); # A126869
c:=n->b(n)/((n/2)+1); # A126120
ch:=n->add(binomial(n, k)*2^(n-k)*c(k)^2, k=0..n); # A202856
[seq(ch(n), n=0..30)];

b[n_] := Coefficient[(x^2+1)^n, x, n]; (* A126869 *)
c[n_] := b[n]/(n/2+1); (* A126120 *)
ch[n_] := Sum[Binomial[n, k] 2^(n-k) c[k]^2, {k, 0, n}]; (* A202856 *)
Table[ch[n], {n, 0, 30}] (* Jean-François Alcover, Aug 10 2018, translated from Maple *)

a(n) = Sum_{k=0..n} binomial(n, k)*2^(n-k)*c(k)^2, where c() = A126120().
Conjecture: (n+2)^2*a(n) +2*(-3*n^2-5*n-1)*a(n-1) -4*(n-1)*(n-5)*a(n-2) +24*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Dec 04 2013 [ Maple's sumrecursion command applied to the above formula for a(n) produces this recurrence. - Peter Bala, Jul 06 2015 ]
a(n) ~ 2^(n-1) * 3^(n+3) / (Pi * n^3). - Vaclav Kotesovec, Jul 20 2019

A138351 Central moment sequence of tr(A^2) in USp(4).

Original entry on oeis.org

1, 0, 2, 1, 11, 16, 95, 232, 1085, 3460, 14820, 54275, 227095, 895688, 3756688, 15462293, 65586405, 277342336, 1192038266, 5136760581, 22357937431, 97730561480, 430177280197, 1901975209706, 8454151507801, 37734802709796

Offset: 0

Andrew V. Sutherland, Mar 16 2008, Mar 31 2008

nonn

If A is a random matrix in the compact group USp(4) (4 X 4 complex matrices which are unitary and symplectic), then a(n) = E[(tr(A^2)+1)^n] is the n-th central moment of the trace of A^2, since E[tr(A^2)] = -1 (see A138350).

			a(4) = 11 because E[(tr(A^2)+1)^4] = 11 for a random matrix A in USp(4).
a(4) = 1*A138350(0)+4*A138350(1)+6*A138350(2)+4*A138350(3)+1*A138350(4) = 1*1 + 4*(-1) + 6*3 + 4*(-6) + 1*20 = 11.

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, in Arithmetic, Geometry, Cryptography, and Coding Theory: International Conference, November 5-9, 2007, CIRM, Marseilles, France. Gilles Lachaud, Christophe Ritzenthaler, Michael A. Tsfasman, editors. 2009. (Contemporary Mathematics ; v.487)..

Cf. A138350.

a126120[n_] := If[EvenQ[n], CatalanNumber[n/2], 0];
a138364[n_] := If[EvenQ[n], 0, Binomial[n, Floor[n/2]], 0];
a138350[n_] := a126120[n] a138364[n+1] - a138364[n] a126120[n+1];
a[n_] := Sum[Binomial[n, i] a138350[i], {i, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 13 2018 *)

a(n) = (1/2)Integral_{x=0..Pi,y=0..Pi}(2cos(2x)+2cos(2y)+1)^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy.

a(n) = Sum_{i=0..n} binomial(n,i)*A138350(i).

cp's OEIS Frontend

A321396 Square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 0, related to a class of Motzkin trees.

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A121448 Triangle read by rows: T(n,k) is the number of binary trees with n edges and having k vertices of outdegree 1 (n>=0, k>=0). A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.

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A138350 Moment sequence of tr(A^2) in USp(4).

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A166135 Number of possible paths to each node that lies along the edge of a cut 4-nomial tree, that is rooted one unit from the cut.

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A222006 Number of forests of rooted plane binary trees (all nodes have outdegree of 0 or 2) with n total nodes.

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A306437 Regular triangle read by rows where T(n,k) is the number of non-crossing set partitions of {1, ..., n} in which all blocks have size k.

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A359364 Triangle read by rows. The Motzkin triangle, the coefficients of the Motzkin polynomials. M(n, k) = binomial(n, k) * CatalanNumber(k/2) if k is even, otherwise 0.

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