A109187 -id:A109187 - cp's OEIS frontend

A089627 T(n,k) = binomial(n,2k)binomial(2*k,k) for 0 <= k <= n, triangle read by rows.

1, 1, 0, 1, 2, 0, 1, 6, 0, 0, 1, 12, 6, 0, 0, 1, 20, 30, 0, 0, 0, 1, 30, 90, 20, 0, 0, 0, 1, 42, 210, 140, 0, 0, 0, 0, 1, 56, 420, 560, 70, 0, 0, 0, 0, 1, 72, 756, 1680, 630, 0, 0, 0, 0, 0, 1, 90, 1260, 4200, 3150, 252, 0, 0, 0, 0, 0, 1, 110, 1980, 9240, 11550, 2772, 0, 0, 0, 0, 0, 0, 1, 132, 2970, 18480, 34650, 16632, 924, 0, 0, 0, 0, 0, 0, 1, 156, 4290, 34320, 90090, 72072, 12012, 0, 0, 0, 0, 0, 0, 0, 1, 182, 6006, 60060, 210210, 252252, 84084, 3432, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 31 2003

The rows of this triangle are the gamma vectors of the n-dimensional type B associahedra (Postnikov et al., p.38 ). Cf. A055151 and A101280. - Peter Bala, Oct 28 2008
T(n,k) is the number of Grand Motzkin paths of length n having exactly k upsteps (1,1). Cf. A109189, A055151. - Geoffrey Critzer, Feb 05 2014
The result Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)*x^k = (sqrt(1 - 4*x))^n* P(n,1/sqrt(1 - 4*x)) expressing the row polynomials of this triangle in terms of the Legendre polynomials P(n,x) is due to Catalan. See Laden, equation 7.10, p. 56. - Peter Bala, Mar 18 2018

			Triangle begins:
  1
  1,   0
  1,   2,    0
  1,   6,    0,     0
  1,  12,    6,     0,     0
  1,  20,   30,     0,     0,     0
  1,  30,   90,    20,     0,     0,   0
  1,  42,  210,   140,     0,     0,   0, 0
  1,  56,  420,   560,    70,     0,   0, 0, 0
  1,  72,  756,  1680,   630,     0,   0, 0, 0, 0
  1,  90, 1260,  4200,  3150,   252,   0, 0, 0, 0, 0
  1, 110, 1980,  9240, 11550,  2772,   0, 0, 0, 0, 0, 0
  1, 132, 2970, 18480, 34650, 16632, 924, 0, 0, 0, 0, 0, 0
Relocating the zeros to be evenly distributed and interpreting the triangle as the coefficients of polynomials
                     1
                     1
                 1 + 2 q^2
                 1 + 6 q^2
            1 + 12 q^2 +  6 q^4
            1 + 20 q^2 + 30 q^4
       1 + 30 q^2 +  90 q^4 +  20 q^6
       1 + 42 q^2 + 210 q^4 + 140 q^6
  1 + 56 q^2 + 420 q^4 + 560 q^6 + 70 q^8
then the substitution q^k -> 1/(floor(k/2)+1) gives the Motzkin numbers A001006.
- _Peter Luschny_, Aug 29 2011

Alois P. Heinz, Rows n = 0..140, flattened
Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), #14.9.1.
Hyman N. Laden, An historical, and critical development of the theory of Legendre polynomials before 1900, Master of Arts Thesis, University of Maryland 1938.
Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013.
A. Postnikov, V. Reiner, and L. Williams, Faces of generalized permutohedra, arXiv:math/0609184 [math.CO], 2006-2007.

Row sums A002426. Antidiagonal sums A098479.

Cf. A055151, A063007, A109187, A101280.

for i from 0 to 12 do seq(binomial(i, j)*binomial(i-j, j), j=0..i) od; # Zerinvary Lajos, Jun 07 2006
# Alternatively:
R := (n, x) -> simplify(hypergeom([1/2 - n/2, -n/2], [1], 4*x)):
Trow := n -> seq(coeff(R(n,x), x, j), j=0..n):
seq(print(Trow(n)), n=0..9); # Peter Luschny, Mar 18 2018

nn=15;mxy=(1-x-(1-2x+x^2-4x^2y)^(1/2))/(2x^2 y);Map[Select[#,#>0&]&, CoefficientList[Series[1/(1-x-2y x^2mxy),{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Feb 05 2014 *)

T(n,k) = binomial(n,2*k)*binomial(2*k,k);
concat(vector(15, n, vector(n, k, T(n-1, k-1)))) \\ Gheorghe Coserea, Sep 01 2018

T(n,k) = n!/((n-2*k)!*k!*k!).
E.g.f.: exp(x)*BesselI(0, 2*x*sqrt(y)). - Vladeta Jovovic, Apr 07 2005
O.g.f.: ( 1 - x - sqrt(1 - 2*x + x^2 - 4*x^2*y))/(2*x^2*y). - Geoffrey Critzer, Feb 05 2014
R(n, x) = hypergeom([1/2 - n/2, -n/2], [1], 4*x) are the row polynomials. - Peter Luschny, Mar 18 2018
From Peter Bala, Jun 23 2023: (Start)
T(n,k) = Sum_{i = 0..k} (-1)^i*binomial(n, i)*binomial(n-i, k-i)^2. Cf. A063007(n,k) = Sum_{i = 0..k} binomial(n, i)^2*binomial(n-i, k-i).
T(n,k) = A063007(n-k,k); that is, the diagonals of this table are the rows of A063007. (End)

A109188 Number of (1,0) steps in all Grand Motzkin paths of length n.

Original entry on oeis.org

1, 2, 9, 28, 95, 306, 987, 3144, 9963, 31390, 98483, 307836, 959257, 2981174, 9243405, 28601712, 88342659, 272428758, 838903371, 2579937060, 7924966749, 24317716038, 74546117121, 228317474952, 698708409525, 2136597743826

Offset: 1

Emeric Deutsch, Jun 21 2005

nonn

A Grand Motzkin path is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).

			a(3)=9 because we have the following 7 (=A002426(3)) Grand Motzkin paths of length 3: hhh, hud, hdu, udh, duh, uhd and dhu; they have a total of 9 h-steps.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A109187, A002426.

g:=z*(1-z)/(1-2*z-3*z^2)^(3/2): gser:=series(g,z=0,33): seq(coeff(gser,z^n),n=1..30);
a := n -> n*hypergeom([1-n/2, 1/2-n/2], [1], 4):
seq(simplify(a(n)), n=1..26); # Peter Luschny, Sep 18 2014

Rest[CoefficientList[Series[x*(1-x)/(1-2*x-3*x^2)^(3/2), {x, 0, 20}], x]] (* Vaclav Kotesovec, Sep 18 2014 *)

Vec(z*(1-z)/(1-2*z-3*z^2)^(3/2) + O(z^50)) \\ G. C. Greubel, Jan 31 2017

G.f.: x*(1 - x)/(1 - 2*x - 3*x^2)^(3/2).
a(n) = n*A002426(n-1). - Paul Barry, Apr 19 2008, corrected Nov 09 2021
E.g.f.: a(n) = n! * [x^n] exp(x)*((1 + x)*BesselI(0, 2*x) + 2*x*BesselI(1, 2*x)). - Peter Luschny, Aug 25 2012
D-finite with recurrence (-n+1)*a(n) + (3*n-4)*a(n-1) + (n+5)*a(n-2) + 3*(-n+2)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012
a(n) = n*hypergeom([1-n/2, 1/2-n/2], [1], 4) . - Peter Luschny, Sep 18 2014
a(n) ~ 3^(n-1/2)*sqrt(n)/(2*sqrt(Pi)). - Vaclav Kotesovec, Sep 18 2014

A171128 A117852*A130595 as lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 9, 3, 1, 19, 28, 18, 4, 1, 51, 95, 70, 30, 5, 1, 141, 306, 285, 140, 45, 6, 1, 393, 987, 1071, 665, 245, 63, 7, 1, 1107, 3144, 3948, 2856, 1330, 392, 84, 8, 1, 3139, 9963, 14148, 11844, 6426, 2394, 588, 108, 9, 1

Offset: 0

Philippe Deléham, Dec 04 2009

Mirror image of triangle in A135091.

Exponential Riordan array [exp(x)*Bessel_I(0,2*x), x] = A007318 * A109187. - Peter Bala, Feb 12 2017

			Triangle begins:
   1
   1  1
   3  2  1
   7  9  3 1
  19 28 18 4 1
  ...
From _Peter Bala_, Feb 12 2017: (Start)
The infinitesimal generator begins
      0
      1    0
      2    2     0
      0    6     3     0
     -6    0    12     4     0
      0  -30     0    20     5   0
     80    0   -90     0    30   6   0
      0  560     0  -210     0  42   7  0
  -2310    0  2240     0  -420   0  56  8  0
  ....
and equals the generalized exponential Riordan array [x + log(Bessel_I(0,2*x)), x], and so has integer entries. (End)

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

A000984 (row sums), A135091 (row reversed). Cf. A002426, A117852, A130595, A109187.

A002426[n_] := Sum[Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}]; Table[ Binomial[n, k]*A002426[n - k], {n, 0, 99}, {k, 0, n} ]//Flatten (* G_. C. Greubel_, Mar 07 2017 *)

Sum_{k=0..n} T(n,k)*x^k = A126869(n), A002426(n), A000984(n), A026375(n), A081671(n), A098409(n), A098410(n), A104454(n) for x = -1,0,1,2,3,4,5,6 respectively.
T(n,k) = binomial(n,k)*A002426(n-k). - Philippe Deléham, Dec 12 2009
From Peter Bala, Feb 12 2017: (Start)
T(n,k) = Sum_{j = 0..floor((n-k)/2)} n!/((n-k-2*j)!*j!^2*k!).
T(n,k) = n/k*T(n-1,k-1) with T(n,0) = A002426(n).
(n - k)^2*T(n,k) = n*(2*n - 2*k - 1)*T(n-1,k) + 3*n*(n - 1)*T(n-2,k).
O.g.f. = 1/sqrt((1 - (1 + t)*z)^2 - 4*z^2) = 1 + (1 + t)*z + (3 + 2*t + t^2)*z^2 + (7 + 9*t + 3*t^2 + t^3 )*z^3 + ....
E.g.f. Bessel_I(0,2*x) * exp((1 + t)*x) = 1 + (1 + t)*z + (3 + 2*t + t^2)*z^2/2! + (7 + 9*t + 3*t^2 + t^3 )*z^3/3! + ....
n-th row polynomial R(n,t) = Sum_{k = 0..floor(n/2)} binomial(n,2*k)*binomial(2*k,k)*(1 + t)^(n-2*k) = coefficient of x^n in the expansion of (1 + (1 + t)*x + x^2)^n.
The polynomials R(n, t - 1) are the row polynomials of A109187.
d/dt(R(n,t)) = n*R(n-1,t).
Moment representation on a finite interval: R(n,t) = 1/Pi * Integral_{x = t-1 .. t+3} x^n/sqrt((t + 3 - x)*(x - t + 1)) dx.
The zeros of the row polynomials appear to lie on the vertical line Re(z) = -1 in the complex plane, and the zeros of R(n,t) and R(n+1,t) appear to interlace along this line.
(End)

A282252 Exponential Riordan array [Bessel_I(0,2*x)^2, x].

Original entry on oeis.org

1, 0, 1, 4, 0, 1, 0, 12, 0, 1, 36, 0, 24, 0, 1, 0, 180, 0, 40, 0, 1, 400, 0, 540, 0, 60, 0, 1, 0, 2800, 0, 1260, 0, 84, 0, 1, 4900, 0, 11200, 0, 2520, 0, 112, 0, 1, 0, 44100, 0, 33600, 0, 4536, 0, 144, 0, 1, 63504, 0, 220500, 0, 84000, 0, 7560, 0, 180, 0, 1

Offset: 0

Peter Bala, Feb 12 2017

Bessel_I(0,2*x) = Sum_{n >= 0} binomial(2*n,n)*x^(2*n)/(2*n)! is a modified Bessel function of the first kind.
Consider the infinite 2-dimensional square lattice Z x Z with an oriented self-loop at each vertex. Then the triangle entry T(n,k) equals the number of walks of length n from the origin to itself having k loops. An example is given below.
See A069466 for walks an infinite 2-dimensional square lattice without self-loops.
This is the square of triangle A109187, whose entries give the number of walks of length n from a vertex to itself having k loops on a 1-dimensional integer lattice with an oriented self-loop at each vertex.
A109187 is the exponential Riordan array [Bessel_I(0,2*x), x]. Note that Bessel_I(0,2*x)^2 = (Sum_{n >= 0} binomial(2*n,n)* x^(2*n)/(2*n)!)^2 = Sum_{n >= 0} binomial(2*n,n)^2*x^(2*n) /(2*n)!.

			The triangle begins
    1;
    0,   1;
    4,   0,   1;
    0,  12,   0,   1;
   36,   0,  24,   0,   1;
    0, 180,   0,  40,   0,   1;
  400,   0, 540,   0,  60,   0,   1;
  ...
T(3,1) = 12: on the square lattice, let L, R, U, D denote a left step, right step, up step and down step respectively. The 12 walks of length 3 containing a single loop are
    loop L R, loop R L, loop U D, loop D U,
    L loop R, R loop L, U loop D, D loop U,
    L R loop, R L loop, U D loop, D U loop.
The infinitesimal generator of this array has integer entries and begins
      0;
      0,    0;
      4,    0,    0;
      0,   12,    0,    0;
    -12,    0,   24,    0,    0;
      0,  -60,    0,   40,    0,    0;
    160,    0, -180,    0,   60,    0,    0;
      0, 1120,    0, -420,    0,   84,    0,    0;
  -4620,    0, 4480,    0, -840,    0,  112,    0,    0;
  ...
It is the generalized exponential Riordan array [ 2*log(Bessel_I(0,2*x)), x ].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

A201805 gives row sums. Cf. A069466, A109187.

T := (n, k) -> (1/2)*binomial(n, k)*binomial(n-k, floor((1/2)*n-(1/2)*k))^2*(1+(-1)^(n-k)):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Table[Binomial[n, k] Binomial[n - k, Floor[(n - k)/2]]^2*(1 + (-1)^(n - k))/2, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 12 2017 *)

for(n=0,10, for(k=0,n, print1(binomial(n,k)*binomial(n-k,floor((n-k)/2))^2*(1 + (-1)^(n-k))/2, ", "))) \\ G. C. Greubel, Aug 16 2017

T(n,k) = binomial(n,k)*binomial(n-k,floor((n-k)/2))^2*(1 + (-1)^(n-k))/2.
T(n,n-2*k) = n/(n - 2*k)*T(n-1,n-2*k-1).
T(n,k) = the coefficient of t^k in the expansion of (t + X + 1/X + Y + 1/Y)^n.
T(n,k) = 1/Pi^2 * Integral_{y = 0..Pi} Integral_{x = 0..Pi} ( t + 2*cos(x) + 2*cos(y) )^n dx dy.
E.g.f.: exp(x*t)*Bessel_I(0,2*x)^2 = 1 + t*x + (4 + t^2)*x^2/2! + (12*t + t^3)*x^3/3! + (36 + 24*t^2 + t^4)*x^4/4! + ....
The n-th row polynomial R(n,t) = Sum_{k = 0..floor(n/2)} binomial(n,2*k)*binomial(2*k,k)^2 * t^(n-2*k).
Recurrence: n^2*R(n,t) = t*(3*n^2 - 3*n + 1)*R(n-1,t) + (16 - 3*t^2)*(n - 1)^2*R(n-2,t) + t*(t^2 - 16)*(n - 1)*(n - 2)*R(n-3,t) with R(0,t) = 1, R(1,t) = t and R(2,t) = 4 + t^2.
d/dt(R(n,t)) = n*R(n-1,t).
The zeros of the row polynomials appear to lie on the imaginary axis in the complex plane. Also, the zeros of R(n,t) and R(n+1,t) appear to interlace on the imaginary axis.

A298608 Polynomials related to the Motzkin numbers for Coxeter type D, T(n, k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 2, 6, 2, 1, 6, 9, 12, 3, 1, 8, 30, 24, 20, 4, 1, 20, 50, 90, 50, 30, 5, 1, 30, 140, 180, 210, 90, 42, 6, 1, 70, 245, 560, 490, 420, 147, 56, 7, 1, 112, 630, 1120, 1680, 1120, 756, 224, 72, 8, 1

Offset: 0

Peter Luschny, Jan 23 2018

The polynomials evaluated at x = 1 give the analog of the Motzkin numbers for Coxeter type D (see A298300 (with a shift in the indexing)).

			The first few polynomials are:
  p0(x) =  1;
  p1(x) =  0 +     x;
  p2(x) =  2 +     x +     x^2;
  p3(x) =  2 +   6*x +   2*x^2 +     x^3;
  p4(x) =  6 +   9*x +  12*x^2 +   3*x^3 +    x^4;
  p5(x) =  8 +  30*x +  24*x^2 +  20*x^3 +   4*x^4 +     x^5;
  p6(x) = 20 +  50*x +  90*x^2 +  50*x^3 +  30*x^4 +   5*x^5 +    x^6;
  p7(x) = 30 + 140*x + 180*x^2 + 210*x^3 +  90*x^4 +  42*x^5 +  6*x^6 +   x^7;
The triangle starts:
  [0][  1]
  [1][  0,   1]
  [2][  2,   1,    1]
  [3][  2,   6,    2,    1]
  [4][  6,   9,   12,    3,    1]
  [5][  8,  30,   24,   20,    4,   1]
  [6][ 20,  50,   90,   50,   30,   5,   1]
  [7][ 30, 140,  180,  210,   90,  42,   6,  1]
  [8][ 70, 245,  560,  490,  420, 147,  56,  7, 1]
  [9][112, 630, 1120, 1680, 1120, 756, 224, 72, 8, 1]

Row sums are A298300(n+1) for n >= 1.

Cf. A109187, A298609.

A298608Poly := n -> `if`(n=0, 1, binomial(2*n, n)*(hypergeom([-n, -n], [-n+1/2], 1/2-x/4) + ((n-1)/(n+1))*hypergeom([-n+1, -n-1], [-n+1/2], 1/2-x/4))):
A298608Row := n -> op(PolynomialTools:-CoefficientList(simplify(A298608Poly(n)), x)): seq(A298608Row(n), n=0..9);

p[0] := 1;
p[n_] := GegenbauerC[n, -n , -x/2] + GegenbauerC[n - 1, -n , -x/2] (n - 1) / n;
Table[CoefficientList[p[n], x], {n, 0, 9}] // Flatten

T(n,k) = A109187(n,k) + A298609(n,k).
The polynomials are defined by p(0, x) = 1 and for n >= 1 by p(n, x) = G(n,-n,-x/2) + G(n-1,-n,-x/2)*(n-1)/n where G(n, a, x) denotes the n-th Gegenbauer polynomial.
p(n, x) = binomial(2*n,n)*(hypergeom([-n,-n], [-n+1/2], 1/2-x/4) + ((n-1)/(n+1))*hypergeom([-n+1,-n-1], [-n+1/2], 1/2-x/4)) for n >= 1.

A298609 Polynomials related to the Motzkin sums for Coxeter type D, T(n, k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 9, 0, 3, 0, 8, 0, 24, 0, 4, 0, 0, 50, 0, 50, 0, 5, 0, 30, 0, 180, 0, 90, 0, 6, 0, 0, 245, 0, 490, 0, 147, 0, 7, 0, 112, 0, 1120, 0, 1120, 0, 224, 0, 8, 0, 0, 1134, 0, 3780, 0, 2268, 0, 324, 0, 9, 0, 420, 0, 6300, 0, 10500, 0, 4200, 0, 450, 0, 10, 0

Offset: 0

Peter Luschny, Jan 23 2018

The polynomials evaluated at x = 1 give the analog of the Motzkin sums for Coxeter type D (see A290380 (with a shift in the indexing)).

			The first few polynomials are:
p0(x) = 0;
p1(x) = 0;
p2(x) = x;
p3(x) = 2 + 2*x^2;
p4(x) = 9*x + 3*x^3;
p5(x) = 8 + 24*x^2 + 4*x^4;
p6(x) = 50*x + 50*x^3 + 5*x^5;
p7(x) = 30 + 180*x^2 + 90*x^4 + 6*x^6;
p8(x) = 245*x + 490*x^3 + 147*x^5 + 7*x^7;
p9(x) = 112 + 1120*x^2 + 1120*x^4 + 224*x^6 + 8*x^8;
The triangle of coefficients extended by the main diagonal with zeros starts:
[0][  0]
[1][  0,    0]
[2][  0,    1,    0]
[3][  2,    0,    2,    0]
[4][  0,    9,    0,    3,    0]
[5][  8,    0,   24,    0,    4,    0]
[6][  0,   50,    0,   50,    0,    5,   0]
[7][ 30,    0,  180,    0,   90,    0,   6,  0]
[8][  0,  245,    0,  490,    0,  147,   0,  7,  0]
[9][112,    0, 1120,    0, 1120,    0, 224,  0,  8,  0]

Cf. A109187, A290380, A298608.

A298609Poly := n -> `if`(n<=1, 0, binomial(2*n, n)*((n-1)/(n+1))*hypergeom([1-n, -n-1], [-n+1/2], 1/2-x/4)):
A298609Row := n -> if n=0 then 0 elif n=1 then 0,0 else op(PolynomialTools:-CoefficientList(simplify(A298609Poly(n)), x)),0 fi:
seq(A298609Row(n), n=0..11);

P298609[n_] := If[n <= 1, 0, GegenbauerC[n - 1, -n, -x/2] (n - 1)/n];
Flatten[ Join[ {{0}, {0, 0}},
  Table[ Join[ CoefficientList[ P298609[n], x], {0}], {n, 2, 10}]]]

A298608(n,k) = A109187(n,k) + T(n,k).
The polynomials are defined by p(0, x) = p(1, x) = 0 and for n >= 2 by  p(n, x) = G(n - 1, -n, -x/2)*(n - 1)/n where G(n, a, x) denotes the n-th Gegenbauer polynomial.
p(n, x) = Catalan(n)*(n-1)*hypergeom([1-n, -n-1], [-n+1/2], 1/2-x/4) for n >= 2.

A298610 Triangle read by rows, the unsigned coefficients of G(n, n, x/2) where G(n,a,x) denotes the n-th Gegenbauer polynomial, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 2, 0, 3, 0, 12, 0, 10, 10, 0, 60, 0, 35, 0, 105, 0, 280, 0, 126, 56, 0, 756, 0, 1260, 0, 462, 0, 840, 0, 4620, 0, 5544, 0, 1716, 330, 0, 7920, 0, 25740, 0, 24024, 0, 6435, 0, 6435, 0, 60060, 0, 135135, 0, 102960, 0, 24310

Offset: 0

Peter Luschny, Jan 25 2018

			[0]   1
[1]   0,    1
[2]   2,    0,    3
[3]   0,   12,    0,    10
[4]  10,    0,   60,     0,    35
[5]   0,  105,    0,   280,     0,    126
[6]  56,    0,  756,     0,  1260,      0,   462
[7]   0,  840,    0,  4620,     0,   5544,     0,   1716
[8] 330,    0, 7920,     0, 25740,      0, 24024,      0, 6435
[9]   0, 6435,    0, 60060,     0, 135135,     0, 102960,    0,  24310

T(2n, 0) = A165817(n). T(n,n) = A088218(n). Row sums are A213684.

Cf. A109187.

with(orthopoly):
seq(seq((-1)^iquo(n-k, 2)*coeff(G(n,n,x/2),x,k), k=0..n), n=0..9);

p[n_] := Binomial[3 n - 1, n] Hypergeometric2F1[-n, 3 n, n + 1/2, 1/2 - x/4];
Flatten[Table[(-1)^Floor[(n-k)/2] Coefficient[p[n], x, k], {n,0,9}, {k,0,n}]]

G(n, x) = binomial(3*n-1, n)*hypergeom([-n, 3*n], [n+1/2], 1/2 - x/4).

cp's OEIS Frontend

A089627 T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.

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A109188 Number of (1,0) steps in all Grand Motzkin paths of length n.

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A171128 A117852*A130595 as lower triangular matrices.

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A282252 Exponential Riordan array [Bessel_I(0,2*x)^2, x].

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A298608 Polynomials related to the Motzkin numbers for Coxeter type D, T(n, k) for n >= 0 and 0 <= k <= n.

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A298609 Polynomials related to the Motzkin sums for Coxeter type D, T(n, k) for n >= 0 and 0 <= k <= n.

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A298610 Triangle read by rows, the unsigned coefficients of G(n, n, x/2) where G(n,a,x) denotes the n-th Gegenbauer polynomial, T(n, k) for 0 <= k <= n.

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A089627 T(n,k) = binomial(n,2k)binomial(2*k,k) for 0 <= k <= n, triangle read by rows.