A122896 -id:A122896 - cp's OEIS frontend

A064189 Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0)=1, T(n,k)=0 if n < k, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1).

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 9, 12, 9, 4, 1, 21, 30, 25, 14, 5, 1, 51, 76, 69, 44, 20, 6, 1, 127, 196, 189, 133, 70, 27, 7, 1, 323, 512, 518, 392, 230, 104, 35, 8, 1, 835, 1353, 1422, 1140, 726, 369, 147, 44, 9, 1, 2188, 3610, 3915, 3288, 2235, 1242, 560, 200, 54, 10, 1

Offset: 0

N. J. A. Sloane, Sep 21 2001

Motzkin triangle read in reverse order.
T(n,k) = number of lattice paths from (0,0) to (n,k), staying weakly above the x-axis and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). Example: T(3,1) = 5 because we have HHU, UDU, HUH, UHH and UUD. Columns 0,1,2 and 3 give A001006 (Motzkin numbers), A002026 (first differences of Motzkin numbers), A005322 and A005323, respectively. - Emeric Deutsch, Feb 29 2004
Riordan array ((1-x-sqrt(1-2x-3x^2))/(2x^2), (1-x-sqrt(1-2x-3x^2))/(2x)). Inverse is the array (1/(1+x+x^2), x/(1+x+x^2)) (A104562). - Paul Barry, Mar 15 2005
Inverse binomial matrix applied to A039598. - Philippe Deléham, Feb 28 2007
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) = T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Mar 27 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
Equals binomial transform of triangle A053121. - Gary W. Adamson, Oct 25 2008
Consider a semi-infinite chessboard with squares labeled (n,k), ranks or rows n >= 0, files or columns k >= 0; the number of king-paths of length n from (0,0) to (n,k), 0 <= k <= n, is T(n,k). The recurrence relation given above relates to the movements of the king. This is essentially the comment made by Harrie Grondijs for the Motzkin triangle A026300. - Johannes W. Meijer, Oct 10 2010

			Triangle begins:
  [0]   1;
  [1]   1,    1;
  [2]   2,    2,    1;
  [3]   4,    5,    3,    1;
  [4]   9,   12,    9,    4,   1;
  [5]  21,   30,   25,   14,   5,   1;
  [6]  51,   76,   69,   44,  20,   6,   1;
  [7] 127,  196,  189,  133,  70,  27,   7,  1;
  [8] 323,  512,  518,  392, 230, 104,  35,  8, 1;
  [9] 835, 1353, 1422, 1140, 726, 369, 147, 44, 9, 1;
  ...
From _Philippe Deléham_, Nov 04 2011: (Start)
Production matrix begins:
  1, 1
  1, 1, 1
  0, 1, 1, 1
  0, 0, 1, 1, 1
  0, 0, 0, 1, 1, 1
  0, 0, 0, 0, 1, 1, 1 (End)

See A026300 for additional references and other information.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
E. Barcucci, R. Pinzani and R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
Paul Barry, Moment sequences, transformations, and Spidernet graphs, arXiv:2307.00098 [math.CO], 2023.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 16, 29.
Xiaomei Chen, Counting humps and peaks in Motzkin paths with height k, arXiv:2412.00668 [math.CO], 2024. See pp. 3, 7.
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.
Igor Dolinka, James East, and Robert D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015.
Robert Donaghey and Louis W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
Ivana Đurđev, Igor Dolinka, and James East, Sandwich semigroups in diagram categories, arXiv:1910.10286 [math.GR], 2019.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Tom Halverson and Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.
Toufik Mansour and Mark Shattuck, Enumeration of Catalan and smooth words according to capacity, Integers (2025) Vol. 25, Art. No. A5. See p. 29.
Donatella Merlini and Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.
Robin Pemantle and Mark C. Wilson, Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, SIAM Rev., 50 (2008), no. 2, 199-272. See p. 265.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.
Sheng-Liang Yang and Yuan-Yuan Gao, The Pascal rhombus and Riordan arrays, Fib. Q., 56:4 (2018), 337-347. See Fig. 3.

A026300 (the main entry for this sequence) with rows reversed.
Cf. A001006, A002026, A005322, A005323, A053121.
Row sums give: A005773(n+1) or A307789(n+2).

alias(C=binomial): A064189 := (n,k) -> add(C(n,j)*(C(n-j,j+k)-C(n-j,j+k+2)), j=0..n): seq(seq(A064189(n,k), k=0..n),n=0..10); # Peter Luschny, Dec 31 2019
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> simplify(hypergeom([1 -n/2, -n/2+1/2], [2], 4))); # Peter Luschny, Oct 08 2022

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, 1, 1], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[(k - n)/2, (k - n + 1)/2, k + 2, 4];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten  (* Peter Luschny, May 19 2021 *)

{T(n, k) = if( k<0 || k>n, 0, polcoeff( polcoeff( 2 / (1 - x + sqrt(1 - 2*x - 3*x^2) - 2*x*y) + x * O(x^n), n), k))}; /* Michael Somos, Jun 06 2016 */

def A064189_triangel(dim):
    M = matrix(ZZ,dim,dim)
    for n in range(dim): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+M[n-1,k]+M[n-1,k+1]
    return M
A064189_triangel(9) # Peter Luschny, Sep 20 2012

Sum_{k=0..n} T(n, k)*(k+1) = 3^n.
Sum_{k=0..n} T(n, k)*T(n, n-k) = T(2*n, n) - T(2*n, n+2)
G.f.: M/(1-t*z*M), where M = 1 + z*M + z^2*M^2 is the g.f. of the Motzkin numbers (A001006). - Emeric Deutsch, Feb 29 2004
Sum_{k>=0} T(m, k)*T(n, k) = A001006(m+n). - Philippe Deléham, Mar 05 2004
Sum_{k>=0} T(n-k, k) = A005043(n+2). - Philippe Deléham, May 31 2005
Column k has e.g.f. exp(x)*(BesselI(k,2*x)-BesselI(k+2,2*x)). - Paul Barry, Feb 16 2006
T(n,k) = Sum_{j=0..n} C(n,j)*(C(n-j,j+k) - C(n-j,j+k+2)). - Paul Barry, Feb 16 2006
n-th row is generated from M^n * V, where M = the infinite tridiagonal matrix with all 1's in the super, main and subdiagonals; and V = the infinite vector [1,0,0,0,...]. E.g., Row 3 = (4, 5, 3, 1), since M^3 * V = [4, 5, 3, 1, 0, 0, 0, ...]. - Gary W. Adamson, Nov 04 2006
T(n,k) = A122896(n+1,k+1). - Philippe Deléham, Apr 21 2007
T(n,k) = (k/n)*Sum_{j=0..n} binomial(n,j)*binomial(j,2*j-n-k). - Vladimir Kruchinin, Feb 12 2011
Sum_{k=0..n} T(n,k)*(-1)^k*(k+1) = (-1)^n. - Werner Schulte, Jul 08 2015
Sum_{k=0..n} T(n,k)*(k+1)^3 = (2*n+1)*3^n. - Werner Schulte, Jul 08 2015
G.f.: 2 / (1 - x + sqrt(1 - 2*x - 3*x^2) - 2*x*y) = Sum_{n >= k >= 0} T(n, k) * x^n * y^k. - Michael Somos, Jun 06 2016
T(n,k) = binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4). - Peter Luschny, May 19 2021
The coefficients of the n-th degree Taylor polynomial of the function (1 - x^2)*(1 + x + x^2)^n expanded about the point x = 0 give the entries in row n in reverse order. - Peter Bala, Sep 06 2022

More terms from Vladeta Jovovic, Sep 23 2001

A005774 Number of directed animals of size n (k=1 column of A038622); number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, where s(0) = 2; also sum of row n+1 of array T in A026323.

Original entry on oeis.org

0, 1, 3, 9, 26, 75, 216, 623, 1800, 5211, 15115, 43923, 127854, 372749, 1088283, 3181545, 9312312, 27287091, 80038449, 234988827, 690513030, 2030695569, 5976418602, 17601021837, 51869858544, 152951628725, 451271872701, 1332147482253

Offset: 0

Simon Plouffe

Number of ordered trees with n+1 edges, having root degree at least 2 and nonroot outdegrees at most 2. - Emeric Deutsch, Aug 02 2002
From Petkovsek's algorithm, this recurrence does not have any closed form solutions. So there is no hypergeometric closed form for a(n). - Herbert S. Wilf
Sum of two consecutive trinomial coefficients starting two positions before central one. Example: a(4) = 10+16 and (1 + x + x^2)^4 = ... + 10*x^2 + 16*x^3 + 19*x^4 + ... - David Callan, Feb 07 2004
Image of n (A001477) under the Motzkin related matrix A107131. Binomial transform of A037952. - Paul Barry, May 12 2005
a(n) = total number of ascents (maximal runs of consecutive upsteps) in all Motzkin (n+1)-paths. For example, the 9 Motzkin 4-paths are FFFF, FFUD, FUDF, FUFD, UDFF, UDUD, UFDF, UFFD, UUDD and they contain a total of 9 ascents and so a(3)=9 (U=upstep, D=downstep, F=flatstep). - David Callan, Aug 16 2006
Image of the sequence (0,1,2,3,3,3,...) under the array A122896. - Paul Barry, Sep 18 2006
This is some kind of Motzkin transform of A079978 because the substitution x-> x*A001006(x) in the independent variable of the g.f. A079978(x) yields 1,0 followed by this sequence here. - R. J. Mathar, Nov 08 2008

			G.f.: x + 3*x^2 + 9*x^3 + 26*x^4 + 75*x^5 + 216*x^6 + 623*x^7 + ...

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
P. Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
D. Gouyou-Beauchamps, G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357.
Christian Krattenthaler, Daniel Yaqubi, Some determinants of path generating functions, II, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265.
Simon Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.

Cf. A038622, A098494, A026010, A005773, A005775, A066822.

a005774 0 = 0
a005774 n = a038622 n 1 -- Reinhard Zumkeller, Feb 26 2013

seq( add(binomial(i,k+1)*binomial(i-k,k), k=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
seq(simplify(GegenbauerC(n-2,-n,-1/2) + GegenbauerC(n-1,-n,-1/2)), n=0..27); # Peter Luschny, May 12 2016

CoefficientList[Series[(1-x-Sqrt[1-2x-3x^2])/(x(1-3x+Sqrt[1-2x-3x^2])),{x,0,30}],x] (* Harvey P. Dale, Sep 20 2011 *)
RecurrenceTable[{a[0]==0, a[1]==1,a[n]==(2n(n+1)a[n-1]+3n(n-1)a[n-2])/ ((n+2)(n-1))},a,{n,30}] (* Harvey P. Dale, Nov 09 2012 *)

s=[0,1]; {A005774(n)=k=(2*(n+2)*(n+1)*s[2]+3*(n+1)*n*s[1])/((n+3)*n); s[1]=s[2]; s[2]=k; k}

{a(n) = if( n<2, n>0, (2 * (n+1) * n *a(n-1) + 3 * (n-1) * n * a(n-2)) / (n+2) / (n-1))}; /* Michael Somos, May 01 2003 */

Inverse binomial transform of [ 0, 1, 5, 21, 84, ... ] (A002054). - John W. Layman
D-finite with recurrence (n+2)*(n-1)*a(n) = 2*n*(n+1)*a(n-1) + 3*n*(n-1)*a(n-2) for all n in Z. - Michael Somos, May 01 2003
E.g.f.: exp(x)*(BesselI(1, 2*x)+BesselI(2, 2*x)). - Vladeta Jovovic, Jan 01 2004
G.f.: (1-x-sqrt(1-2x-3x^2))/(x(1-3x+sqrt(1-2x-3x^2))); a(n)= Sum_{k=0..n} C(k+1, n-k+1)*C(n, k)*k/(k+1); a(n) = Sum_{k=0..n} C(n, k)*C(k, floor((k-1)/2)). - Paul Barry, May 12 2005
Starting (1, 3, 9, 26, ...) = binomial transform of A026010: (1, 2, 4, 7, 14, 25, 50, 91, ...). - Gary W. Adamson, Oct 22 2007
a(n)*(2+n) = (4+4*n)*a(n-1) - n*a(n-2) + (12-6*n)*a(n-3). - Simon Plouffe, Feb 09 2012
a(n) ~ 3^(n+1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Mar 10 2014
0 = a(n)*(+36*a(n+1) + 18*a(n+2) - 96*a(n+3) + 30*a(n+4)) + a(n+1)*(-6*a(n+1) + 49*a(n+2) - 26*a(n+3) + 3*a(n+4)) + a(n+2)*(+15*a(n+3) - 8*a(n+4)) + a(n+3)*(a(n+4)) if n >= 0. - Michael Somos, Aug 06 2014
a(n) = GegenbauerC(n-2,-n,-1/2) + GegenbauerC(n-1,-n,-1/2). - Peter Luschny, May 12 2016

Further descriptions from Clark Kimberling

A349812 Triangle read by rows: row 1 is [1]; for n >= 1, row n gives coefficients of expansion of (-1/x + x)*(1/x + 1 + x)^(n-1) in order of increasing powers of x.

Original entry on oeis.org

1, -1, 0, 1, -1, -1, 0, 1, 1, -1, -2, -2, 0, 2, 2, 1, -1, -3, -5, -4, 0, 4, 5, 3, 1, -1, -4, -9, -12, -9, 0, 9, 12, 9, 4, 1, -1, -5, -14, -25, -30, -21, 0, 21, 30, 25, 14, 5, 1, -1, -6, -20, -44, -69, -76, -51, 0, 51, 76, 69, 44, 20, 6, 1, -1, -7, -27, -70, -133, -189, -196, -127, 0, 127, 196, 189, 133, 70, 27, 7, 1

Offset: 0

N. J. A. Sloane, Dec 23 2021

The rule for constructing this triangle (ignoring row 0) is the same as that for A027907: each number is the sum of the three numbers immediately above it in the previous row. Here row 1 is [-1, 0, 1] instead of [1, 1, 1].

			Triangle begins:
   1;
  -1,  0,   1;
  -1, -1,   0,   1,    1;
  -1, -2,  -2,   0,    2,    2,    1;
  -1, -3,  -5,  -4,    0,    4,    5,    3,  1;
  -1, -4,  -9, -12,   -9,    0,    9,   12,  9,   4,   1;
  -1, -5, -14, -25,  -30,  -21,    0,   21, 30,  25,  14,   5,   1;
  -1, -6, -20, -44,  -69,  -76,  -51,    0, 51,  76,  69,  44,  20,  6,  1;
  -1, -7, -27, -70, -133, -189, -196, -127,  0, 127, 196, 189, 133, 70, 27, 7, 1;
  ...

Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.]

Cf. A007318, A027907, A112467, A349813, A348815.

The left half of the triangle is A026300, the right half is A064189 (or A122896). The central (nonzero) column gives the Motzkin numbers A001006.

t1:=-1/x+x; m:=1/x+1+x;
lprint([1]);
for n from 1 to 12 do
w1:=expand(t1*m^(n-1));
w3:=expand(x^n*w1);
w4:=series(w3,x,2*n+1);
w5:=seriestolist(w4);
lprint(w5);
od:

A383527 Partial sums of A005773.

Original entry on oeis.org

1, 2, 4, 9, 22, 57, 153, 420, 1170, 3293, 9339, 26642, 76363, 219728, 634312, 1836229, 5328346, 15494125, 45137995, 131712826, 384900937, 1126265986, 3299509114, 9676690939, 28407473191, 83470059532, 245465090758, 722406781935, 2127562036990, 6270020029353

Offset: 0

Mélika Tebni, Apr 29 2025

For p prime of the form 4*k+3 (A002145), a(p) == 0 (mod p).
For p Pythagorean prime (A002144), a(p) - 2 == 0 (mod p).
a(n) (mod 2) = A010059(n).
a(A000069(n+1)) is even.
a(A001969(n+1)) is odd.

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A000069, A001969, A002144, A002145, A005773, A005774, A005775, A010059, A097893, A122896, A167630, A210736, A211278, A257520.

gf := (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)):
a := n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n = 0 .. 29);
# Recurrence:
a:= proc(n) option remember; `if`(n<=2, 2^n, 3*a(n-1) - (6/n-1)*a(n-2) + (6/n-3)*a(n-3)) end:
seq(a(n), n = 0 .. 29);

Module[{a, n}, RecurrenceTable[{a[n] == 3*a[n-1] - (6-n)*a[n-2]/n + 3*(2-n)*a[n-3]/n, a[0] == 1, a[1] == 2, a[2] == 4}, a, {n, 0, 30}]] (* Paolo Xausa, May 05 2025 *)

from math import comb as C
def a(n):
  return sum(C(n, k)*abs(sum((-1)**j*C(k, j) for j in range(k//2 + 1))) for k in range(n + 1))
print([a(n) for n in range(30)])

First differences of A211278.
a(n) = Sum_{k=0..n} A167630(n, k).
Binomial transform of A210736 (see Python program).
G.f.: (1 + sqrt((1 + x) / (1 - 3*x))) / (2*(1 - x)).
E.g.f.: (Integral_{x=-oo..oo} BesselI(0,2*x) dx + (1 + BesselI(0,2*x)) / 2)*exp(x).
Recurrence: n*a(n) = 3*n*a(n-1) - (6-n)*a(n-2) + 3*(2-n)*a(n-3). If n <= 2, a(n) = 2^n.
a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, May 02 2025
From Mélika Tebni, May 09 2025: (Start)
a(n) = A257520(n) + A097893(n-1) for n > 0.
a(n) = Sum_{j=0..n}(Sum_{k=0..j} A122896(j, k)).
a(n+2) - 3*a(n+1) + 2*a(n) = A005774(n).
a(n+2) - 4*a(n+1) + 4*a(n) - a(n-1) = A005775(n) for n >= 3. (End)

A122897 Riordan array (1/(1-x), c(x)-1) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 5, 1, 1, 22, 19, 7, 1, 1, 64, 67, 34, 9, 1, 1, 196, 232, 144, 53, 11, 1, 1, 625, 804, 573, 261, 76, 13, 1, 1, 2055, 2806, 2211, 1171, 426, 103, 15, 1, 1, 6917, 9878, 8399

Offset: 0

Paul Barry, Sep 18 2006

Product of A007318 and A122896. Inverse of Riordan array ((1+x+x^2)/(1+x)^2,x/(1+x)^2). Row sums are A024718.

The n-th row polynomial (in descending powers of x) equals the n-th Taylor polynomial of the rational function (1 - x^2)/(1 + x + x^2) * (1 + x)^(2*n) about 0. For example, for n = 4 we have (1 - x^2)/( 1 + x + x^2) * (1 + x)^8 = (x^4 + 22*x^3 + 19*x^2 + 7*x + 1) + O(x^5). - Peter Bala, Feb 21 2018

			Triangle begins
  1,
  1,     1,
  1,     3,     1,
  1,     8,     5,     1,
  1,    22,    19,     7,     1,
  1,    64,    67,    34,     9,    1,
  1,   196,   232,   144,    53,   11,    1,
  1,   625,   804,   573,   261,   76,   13,   1,
  1,  2055,  2806,  2211,  1171,  426,  103,  15,   1,
  1,  6917,  9878,  8399,  4979, 2126,  647, 134,  17,  1,
  1, 23713, 35072, 31655, 20483, 9878, 3554, 932, 169, 19, 1

P. Bala, A 4-parameter family of embedded Riordan arrays

A122897 := proc (n, k)
  binomial(2*n, n-k) + 2*add(cos((2/3)*Pi*j)*binomial(2*n, n-k-j), j = 1..n-k)
end proc:
for n from 0 to 10 do
seq(A122897(n, k), k = 0..n)
end do; # Peter Bala, Feb 21 2018

T(n,k) = binomial(2*n,n-k) + 2*Sum_{j = 1..n-k} cos((2/3)*Pi*j)* binomial(2*n, n-k-j). - Peter Bala, Feb 21 2018

T(n,k) = k*Sum_{i=0..n-k} C(2*(i+k),i)/(i+k), T(n,0)=1. - Vladimir Kruchinin, Jun 13 2020

cp's OEIS Frontend

A064189 Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0)=1, T(n,k)=0 if n < k, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1).

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A005774 Number of directed animals of size n (k=1 column of A038622); number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, where s(0) = 2; also sum of row n+1 of array T in A026323.

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A349812 Triangle read by rows: row 1 is [1]; for n >= 1, row n gives coefficients of expansion of (-1/x + x)*(1/x + 1 + x)^(n-1) in order of increasing powers of x.

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A383527 Partial sums of A005773.

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A122897 Riordan array (1/(1-x), c(x)-1) where c(x) is the g.f. of A000108.

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A344566 T(n, k) = (-1)^(n - k)*binomial(n - 1, k - 1)*hypergeom([-(n - k)/2, -(n - k - 1)/2], [1 - n], 4). Triangle read by rows, T(n, k) for 0 <= k <= n.

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A344566 T(n, k) = (-1)^(n - k)binomial(n - 1, k - 1)hypergeom([-(n - k)/2, -(n - k - 1)/2], [1 - n], 4). Triangle read by rows, T(n, k) for 0 <= k <= n.