A126954 -id:A126954 - cp's OEIS frontend

A126791 Binomial matrix applied to A111418.

1, 4, 1, 17, 7, 1, 75, 39, 10, 1, 339, 202, 70, 13, 1, 1558, 1015, 425, 110, 16, 1, 7247, 5028, 2400, 771, 159, 19, 1, 34016, 24731, 12999, 4872, 1267, 217, 22, 1, 160795, 121208, 68600, 28882, 8890, 1940, 284, 25, 1, 764388, 593019, 355890, 164136

Offset: 0

Philippe Deléham, Mar 14 2007

Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 4*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k >= 1.
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
From R. J. Mathar, Mar 12 2013: (Start)
The matrix inverse starts
      1;
     -4,     1;
     11,    -7,     1;
    -29,    31,   -10,    1;
     76,  -115,    60,  -13,    1;
   -199,   390,  -285,   98,  -16,   1;
    521, -1254,  1185, -566,  145, -19,   1;
  -1364,  3893, -4524, 2785, -985, 201, -22, 1; ... (End)

			Triangle begins:
      1;
      4,     1;
     17,     7,     1;
     75,    39,    10,    1;
    339,   202,    70,   13,    1;
   1558,  1015,   425,  110,   16,   1;
   7247,  5028,  2400,  771,  159,  19,  1;
  34016, 24731, 12999, 4872, 1267, 217, 22, 1; ...
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins:
  4, 1
  1, 3, 1
  0, 1, 3, 1
  0, 0, 1, 3, 1
  0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 0, 0, 1, 3, 1 (End)

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

A126791 := proc(n,k)
    if n=0 and k = 0 then
        1 ;
    elif k <0 or k>n then
        0;
    elif k= 0 then
        4*procname(n-1,0)+procname(n-1,1) ;
    else
        procname(n-1,k-1)+3*procname(n-1,k)+procname(n-1,k+1) ;
    end if;
end proc: # R. J. Mathar, Mar 12 2013
T := (n,k) -> (-1)^(n-k)*simplify(GegenbauerC(n-k,-n+1,3/2) - GegenbauerC(n-k-1, -n+1, 3/2)): seq(seq(T(n,k),k=1..n),n=1..10); # Peter Luschny, May 13 2016

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,
T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 4, 3], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A026378(m+n+1).
Sum_{k=0..n} T(n,k) = 5^n = A000351(n).
T(n,k) = (-1)^(n-k)*(GegenbauerC(n-k,-n+1,3/2) - GegenbauerC(n-k-1,-n+1,3/2)). - Peter Luschny, May 13 2016
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 + x )*(1 + 3*x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022

A126953 Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k+1) for k >= 1.

Original entry on oeis.org

1, 3, 1, 10, 3, 1, 33, 11, 3, 1, 110, 36, 12, 3, 1, 366, 122, 39, 13, 3, 1, 1220, 405, 135, 42, 14, 3, 1, 4065, 1355, 447, 149, 45, 15, 3, 1, 13550, 4512, 1504, 492, 164, 48, 16, 3, 1, 45162, 15054, 5004, 1668, 540, 180, 51, 17, 3, 1

Offset: 0

Philippe Deléham, Mar 19 2007

This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007

Riordan array (2/(1-6x+sqrt(1-4*x^2)),x*c(x^2)) where c(x)= g.f. of the Catalan numbers A000108. - Philippe Deléham, Jun 01 2013

			Triangle begins:
     1;
     3,    1;
    10,    3,   1;
    33,   11,   3,   1;
   110,   36,  12,   3,  1;
   366,  122,  39,  13,  3,  1;
  1220,  405, 135,  42, 14,  3, 1;
  4065, 1355, 447, 149, 45, 15, 3, 1;

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

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 3, 0], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)

Sum_{k=0..n} T(n,k) = A127359(n).
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A126931(m+n).
Sum_{k=0..n} T(n,k)*(-2*k+1) = 2^n. - Philippe Deléham, Mar 25 2007

A059738 Binomial transform of A054341 and inverse binomial transform of A049027.

Original entry on oeis.org

1, 3, 10, 34, 117, 405, 1407, 4899, 17083, 59629, 208284, 727900, 2544751, 8898873, 31125138, 108881166, 380928795, 1332824049, 4663705782, 16319702046, 57109857519, 199859075307, 699435489795, 2447823832671, 8566818534141, 29982268505595, 104933418068332

Offset: 0

John W. Layman, Feb 09 2001

nonn

First column of the Riordan array ((1-2x)/(1+x+x^2),x/(1+x+x^2))^(-1). [Paul Barry, Nov 06 2008]
Apparently the Motzkin transform of A125176, supposed A125176 is interpreted with offset 0. [R. J. Mathar, Dec 11 2008]
a(n) is the number of Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors. Example: a(3)=34 because, denoting  U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH, 3 paths of shape HUD, 3 paths of shape UDH, and 1 path of shape UHD. - Emeric Deutsch, May 02 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Taras Goy and Mark Shattuck, Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Table[SeriesCoefficient[2/(1-5*x+Sqrt[1-2*x-3*x^2]),{x,0,n}],{n,0,20}]

x='x+O('x^66); Vec(2/(1-5*x+sqrt(1-2*x-3*x^2))) \\ Joerg Arndt, May 06 2013

a(n) = Sum[k=0..n, 2^(n-k)*A026300(n, k) ], where A026300 is the Motzkin triangle. - Ralf Stephan, Jan 25 2005 [Corrected by Philippe Deléham, Nov 29 2009]
a(n)= A126954(n,0). [Philippe Deléham, Nov 24 2009]
G.f.: 2/(1-5*x+sqrt(1-2*x-3*x^2)). - Emeric Deutsch, May 02 2011
Recurrence: 2*(n+1)*a(n) = (11*n+5)*a(n-1) - (8*n+5)*a(n-2) - 21*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 11 2012
a(n) ~ 3*7^n/2^(n+2). - Vaclav Kotesovec, Oct 11 2012
G.f.: 1/(1 - 3*x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 19 2021

More terms from Vincenzo Librandi, May 06 2013

A097357 For definition see Comments lines.

Original entry on oeis.org

1, 2, 1, 3, 3, 4, 3, 6, 3, 6, 3, 7, 5, 8, 5, 11, 3, 6, 3, 9, 9, 12, 9, 16, 5, 10, 5, 13, 11, 16, 11, 22, 3, 6, 3, 9, 9, 12, 9, 18, 9, 18, 9, 21, 15, 24, 15, 31, 5, 10, 5, 15, 15, 20, 15, 28, 11, 22, 11, 27, 21, 32, 21, 43, 3, 6, 3, 9, 9, 12, 9, 18, 9, 18, 9, 21, 15, 24, 15, 33, 9, 18, 9, 27, 27

Offset: 1

Creighton Dement, Aug 08 2004

nonn

Let b_n(m) represent the m-th entry of the n-th sequence (n > 0) of some family, with the following properties: (b_1(m)) = (0,1,0,0,0,0,0,0,0,0,...) (first term of sequence is m = 0 -> b_1(1)=1 ). For all m, n in naturals ( > 0 ):
Rule I: m > n > 0 -> b_n(m) = 0.
Rule II: b_n(n) = 1.
Rule III: |b_n(m+1) - b_n(m-1)| = 1 -> b_(n+1)(m) = 1 if b_n(m) = 0; b_(n+1)(m) = 0 if b_n(m) = 1; otherwise (i.e., |b_n(m+1) - b_n(m-1)| != 1 -> |b_n(m+1) - b_n(m-1)| = 0) b_(n+1)(m) = b_n(m).
Rule IV: b_n(0) = 0 (this is so that rule III can still be applied to b_n(1)).
The sequence (a(n)) = (a(1), a(2), ...) is then given by a(n) = Sum_{i>=0} b_n(i) = Sum_{i=1..n} b_n(i).
The sequence can be visualized as certain interactions between concentric rings.
This sequence may be connected with Sierpinski's triangle. Details of this as well as a visualization of the rules of "interaction" are given at the link. It is not currently known if this sequence is bounded. The various aligned "triangles of zeros" (apparently each with a number of rows equal to a factor of 8) one sees when using the computer program alude to Sierpinski's Triangle.
At certain points one notices that adjacent terms are all divisible by a certain number -- if this number is divided out one gets back initial terms of the sequence. For example, observe the subsequence (second line, above): 3,6,3,9,9,12,9,18,9,18,9,21,15,24,15,31,5,10,5,15,15,20,15,28,11,22,11,27, divide the first 15 terms by 3 -> 1,2,1,3,3,4,3,6,3,6,3,7,5,8,5 (this is the beginning of the sequence). Skip the number 31 and divide the next 7 terms by 5 -> (1,2,1,3,3,4,3). As the sequence gets longer, it apparently begins repeating (by some factor) an ever-increasing number of its initial terms; for example, another subsequence is: 3,6,3,9,9,12,9,18,9,18,9,21,15,24,15,33,9,18,9,27,27,36,27,48,15,30 = 3*(1,2,1,3,3,4,3,6,3,6,3,7,5,8,5,11,3,6,3,9,9,12,9,16,5,10).

			From _Philippe Deléham_, Apr 28 2007: (Start)
Table b_n(m), n >= 1, m >= 0, begins:
  0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
  0, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...
  0, 0, 0, 1, 0, 0, 0, 0, 0, 0, ...
  0, 0, 1, 1, 1, 0, 0, 0, 0, 0, ...
  0, 1, 0, 1, 0, 1, 0, 0, 0, 0, ...
  0, 1, 0, 1, 0, 1, 1, 0, 0, 0, ...
  0, 1, 0, 1, 0, 0, 0, 1, 0, 0, ...
  0, 1, 0, 1, 1, 0, 1, 1, 1, 0, ...
See A128810 for another version. (End)

// A simple Java program is given at the link provided.

a(n+1) = Sum_{k=0..n} (T(n,k) mod 2) where T = A026300(Motzkin triangle), A064189, A084536, A091965, A110877, A125906, A126954. - Philippe Deléham, Apr 28 2007

A128810 Triangle formed by reading triangle A064189 mod 2 .

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1

Offset: 0

Philippe Deléham, Apr 09 2007

Also triangle formed by reading triangles A091965, A108149, A110877, A125906, A126954 mod 2 .

			Triangle begins:
1;
1, 1;
0, 0, 1;
0, 1, 1, 1;
1, 0, 1, 0, 1;
1, 0, 1, 0, 1, 1;
1, 0, 1, 0, 0, 0, 1;
1, 0, 1, 1, 0, 1, 1, 1;
1, 0, 0, 0, 0, 0, 1, 0, 1;
1, 1, 0, 0, 0, 1, 1, 0, 1, 1;
0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1 ;...

Sum_{k, 0<=k<=n}T(n,k)*x^k=A039963(n), A097357(n+1), A110565(n+1) for x=0,1,2 respectively . T(n,k)= (T(n-1,k-1)+T(n-1,k)+T(n-1,k+1)) mod 2, T(0,0)=1, T(n,k)=0 if k<0 or if k>n .

A301475 Triangular array of polynomials related to the Motzkin triangle and to rooted polyominoes, coefficients in ascending order, read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 4, 5, 3, 1, 5, 3, 1, 3, 1, 1, 9, 12, 9, 4, 1, 12, 9, 4, 1, 9, 4, 1, 4, 1, 1, 21, 30, 25, 14, 5, 1, 30, 25, 14, 5, 1, 25, 14, 5, 1, 14, 5, 1, 5, 1, 1, 51, 76, 69, 44, 20, 6, 1, 76, 69, 44, 20, 6, 1, 69, 44, 20, 6, 1, 44, 20, 6, 1, 20, 6, 1, 6, 1, 1

Offset: 0

Peter Luschny, Mar 22 2018

Evaluating this triangle of polynomials at different values of x leads to interesting integer triangles. For instance at x = 0 it gives the Motzkin triangle A064189 (A026300), at x = 1 it counts rooted polyominoes A038622; at x = 2 it gives A126954 and at x =-1 gives A089942; x = 1/2 and scaling gives A301477.

			Triangle of polynomials starts:
                                    1
                                 1 + x, 1
                          2 + 2 x + x^2, 2 + x, 1
               4 + 5 x + 3 x^2  + x^3, 5 + 3 x^2 + x, 3 + x, 1
9 + 12 x + 9 x^2  + 4 x^3  + x^4, 12 + 9 x + 4 x^2 + x^3, 9 + 4 x + x^2, 4 + x, 1
.
Triangle of coefficients starts:
                               1
                            1, 1, 1
                        2, 2, 1, 2, 1, 1
                  4, 5, 3, 1, 5, 3, 1, 3, 1, 1
         9, 12, 9, 4, 1, 12, 9, 4, 1, 9, 4, 1, 4, 1, 1
21, 30, 25, 14, 5, 1, 30, 25, 14, 5, 1, 25, 14, 5, 1, 14, 5, 1, 5, 1, 1

Cf. A064189 (A026300) (x=0), A038622 (x=1), A126954 (x=2), A089942 (x=-1), A301477 (x=1/2, scaled).

Cf. A000244 (row sums), A000217 (row length).

CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
T := (n,k) -> binomial(n,k)*hypergeom([-k/2,1/2-k/2], [-k+n+2], 4);
P := (n,m) -> add(simplify(T(n,k)*x^(n-k-m)), k=0..n-m);
for n from 0 to 5 do seq(sort(P(n,j),x,ascending), j=0..n) od;
for n from 0 to 5 do seq(CoeffList(P(n,j)), j=0..n) od;

P(n,k) = Sum_{j=0..n-k}binomial(n,j)*hypergeom([-j/2,1/2-j/2],[n-j+2],4)*x^(n-j-k).
T(n,k) is the list of the coefficients of P(n,k) in ascending order.
Row sums are powers of 3, row lengths are the triangular numbers.

A344567 A(n, k) = [x^k] 2 / (1 - (2n - 1)x + sqrt(1 - 2x - 3x^2)). The number of n-colored Motzkin arcs of length k. Array read by ascending antidiagonals, n >= 0 and k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 4, 3, 1, 4, 10, 13, 9, 6, 1, 5, 17, 34, 35, 21, 15, 1, 6, 26, 73, 117, 96, 51, 36, 1, 7, 37, 136, 315, 405, 267, 127, 91, 1, 8, 50, 229, 713, 1362, 1407, 750, 323, 232, 1, 9, 65, 358, 1419, 3741, 5895, 4899, 2123, 835, 603

Offset: 0

Peter Luschny, May 24 2021

Given a sequence a(n), we call the sequence b(n) Cameron's inverse of a, or, as dubbed by Sloane, INVERTi(a) (see the link 'Transforms' in the footer of the page), if 1 + Sum_{n>=1} a(n)*x^n = 1/(1 - Sum_{n>=1} b(n)*x^n).
Iterating this transform starting from A344506 we get:
  a = A344506.
  INVERTi(a) = A059738.
  INVERTi(INVERTi(a)) = A005773.
  INVERTi(INVERTi(INVERTi(a))) = A001006, Motzkin numbers.
  INVERTi(INVERTi(INVERTi(INVERTi(a)))) = A005043.
  INVERTi(INVERTi(INVERTi(INVERTi(INVERTi(a))))) = A344507.
The sequences generated in this manner correspond to the evaluation of the Motzkin polynomials (coefficients in A064189) at x = 3, 2, 1, 0, -1, -2. In terms of ordinary generating functions we have a ZZ-indexed sequence of sequences which general form is given by the formula in the name.
A "Motzkin path of length n and height k" is an integer lattice path from (0, 0) to (n, k) remaining weakly above the x-axis and consisting of steps in {U, L, D}. These acronyms stand for the steps Up = (1,1), Level = (1,0), and Down = (1, -1). An "n-colored Motzkin arc of length k" is a Motzkin path of length k and height 0 where each Level step of height 0 has one of n colors. A(n, k) is the number of n-colored Motzkin arcs of length k. The Motzkin numbers are M(k) = A(1, k).

			Array begins at n = 0, row for n = -1 added for illustration:
n\k  0   1    2     3      4       5        6         7  ... [Sequence Triangle]
--------------------------------------------------------------------------------
[-1] 1, -1,   2,   -2,     5,     -3,      15,        3, ...  [A344507]
[ 0] 1,  0,   1,    1,     3,      6,      15,       36, ...  [A005043, A089942]
[ 1] 1,  1,   2,    4,     9,     21,      51,      127, ...  [A001006, A064189]
[ 2] 1,  2,   5,   13,    35,     96,     267,      750, ...  [A005773, A038622]
[ 3] 1,  3,  10,   34,   117,    405,    1407,     4899, ...  [A059738, A126954]
[ 4] 1,  4,  17,   73,   315,   1362,    5895,    25528, ...  [A344506]
[ 5] 1,  5,  26,  136,   713,   3741,   19635,   103071, ...
[ 6] 1,  6,  37,  229,  1419,   8796,   54531,   338082, ...
[ 7] 1,  7,  50,  358,  2565,  18381,  131727,   944035, ...
[ 8] 1,  8,  65,  529,  4307,  35070,  285567,  2325324, ...
.
Triangle starts:
[0] 1;
[1] 1, 0;
[2] 1, 1,  1;
[3] 1, 2,  2,   1;
[4] 1, 3,  5,   4,   3;
[5] 1, 4, 10,  13,   9,    6;
[6] 1, 5, 17,  34,  35,   21,   15;
[7] 1, 6, 26,  73, 117,   96,   51,  36;
[8] 1, 7, 37, 136, 315,  405,  267, 127,  91;
[9] 1, 8, 50, 229, 713, 1362, 1407, 750, 323, 232.
.
Number of colors = 2, length = 4  ->  35.
.
      /\      _ _
     /  \    /   \   /\/\      3 x 1
.    _         _
    / \_     _/ \              2 x 2
.
    /\_ _   _ _/\   _/\_       3 x 4
.
    _ _ _ _                    1 x 16
.
Number of colors = 4, length = 2  ->  17.
.
    /\                         1 x 1
.
    _ _                        1 x 16

Martin Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discrete Mathematics, Vol. 204, No. 1-3 (1999), 73-112.
Isaac DeJager, Madeleine Naquin, Frank Seidl, Paul Drube, Colored Motzkin Paths of Higher Order, Journal of Integer Sequences, Vol. 24 (2021)

Cf. A064189, A097609, A344506, A059738, A005773, A001006, A005043, A344507.

Cf. triangles: A089942, A064189, A038622, A126954.

Arow := proc(n, len) option remember;
2 / (1 - (2*n - 1)*x + sqrt(1 - 2*x - 3*x^2));
seq(coeff(series(%, x, len+2), x, k), k = 0..len) end:
T := (n, k) -> Arow(n-k, k+1)[k+1]:
for n from 0 to 9 do Arow(n, 7) od; # prints array
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # prints triangle
# Alternative via series reversion:
for n from -1 to 6 do  # print the array starting from n = -1
rgf := x*((n - 1)*x + 1) / ((n^2 - n + 1)*x^2 + (2*n - 1)*x + 1):
subsop(1 = NULL, gfun:-seriestolist(series(rgf, x, 18), 'revogf')) od;
# Via recursively defined polynomials:
p := proc(n, k) option remember;
if n = k then 1 elif k < 0 or n < 0 or k > n then 0 elif k = 0 then x*p(n-1, 0) + p(n-1, 1) else p(n-1, k-1) + p(n-1, k) + p(n-1, k+1) fi end:
A := (n, k) -> subs(x = n, p(k, 0)):
for n from 0 to 8 do lprint(seq(A(n, k), k = 0..9)) od;
# Computing the columns:
Acol := proc(k, len) seq(subs(x = n, p(k, 0)), n = 0..len) end:
for k from 0 to 6 do Acol(k, 9) od;

Unprotect[Power]; 0^0 := 1;
A[n_, k_] := Sum[(n-1)^j Binomial[k, j] Hypergeometric2F1[(j - k)/2, (j - k + 1)/2, j + 2, 4], {j, 0, k}]; Table[A[n, k], {n, 0, 6}, {k, 0, 8}]

F(n) = {x*((n - 1)*x + 1) / ((n^2 - n + 1)*x^2 + (2*n - 1)*x + 1)}
M(n,m=n) = {Mat(vectorv(n, i, Vec(serreverse(F(i-1) + O(x*x^m)))))}
{ my(A=M(8)); for(n=1, #A, print(A[n, ])) } \\ Andrew Howroyd, May 27 2021

def Arow(n, len):
    R. = PowerSeriesRing(QQ, default_prec=len)
    f = x*((n - 1)*x + 1) / ((n^2 - n + 1)*x^2 + (2*n - 1)*x + 1)
    return f.reverse().shift(-1).list()
for n in (0..8): print(Arow(n,10))

A(n, k) = Sum_{j=0..n} (k - 1)^j*binomial(n, j)*hypergeom([(j - n)/2, (j - n + 1)/2], [j + 2], 4).
Arow(n) = [x^n] reverse(x*((n-1)*x + 1) / ((n^2 - n + 1)*x^2 + (2*n-1)*x + 1)) / x.
Computationally more elementary is the following procedure: Let P_n(x) be polynomials defined recursively by P_n(x) = p(n, 0) where p(n, k) = 0 if k < 0 or n < 0 or k > n, p(n, n) = 1, p(n, 0) = x*p(n-1, 0) + p(n-1, 1), and in all other cases p(n, k) = p(n-1, k-1) + p(n-1, k) + p(n-1, k+1). Then A(n, k) = P_k(n).
The coefficients of these polynomials are in A097609. Thus the columns of the array can be calculated as: Acol(k) = [P_k(n) for n >= 0].

cp's OEIS Frontend

A126791 Binomial matrix applied to A111418.

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A126953 Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k+1) for k >= 1.

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A059738 Binomial transform of A054341 and inverse binomial transform of A049027.

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A097357 For definition see Comments lines.

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A128810 Triangle formed by reading triangle A064189 mod 2 .

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A301475 Triangular array of polynomials related to the Motzkin triangle and to rooted polyominoes, coefficients in ascending order, read by rows, for 0 <= k <= n.

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A344567 A(n, k) = [x^k] 2 / (1 - (2*n - 1)*x + sqrt(1 - 2*x - 3*x^2)). The number of n-colored Motzkin arcs of length k. Array read by ascending antidiagonals, n >= 0 and k >= 0.

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A344567 A(n, k) = [x^k] 2 / (1 - (2n - 1)x + sqrt(1 - 2x - 3x^2)). The number of n-colored Motzkin arcs of length k. Array read by ascending antidiagonals, n >= 0 and k >= 0.