A005323 -id:A005323 - 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

A026300 Motzkin triangle, T, read by rows; T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

Right-hand columns have g.f. M^k, where M is g.f. of Motzkin numbers.

Consider a semi-infinite chessboard with squares labeled (n,k), ranks or rows n >= 0, files or columns k >= 0; number of king-paths of length n from (0,0) to (n,k), 0 <= k <= n, is T(n,n-k). - Harrie Grondijs, May 27 2005. Cf. A114929, A111808, A114972.

			Triangle starts:
  [0] 1;
  [1] 1, 1;
  [2] 1, 2,  2;
  [3] 1, 3,  5,   4;
  [4] 1, 4,  9,  12,   9;
  [5] 1, 5, 14,  25,  30,  21;
  [6] 1, 6, 20,  44,  69,  76,   51;
  [7] 1, 7, 27,  70, 133, 189,  196,  127;
  [8] 1, 8, 35, 104, 230, 392,  518,  512,  323;
  [9] 1, 9, 44, 147, 369, 726, 1140, 1422, 1353, 835.

Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.
A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675.
Tewodros Amdeberhan, Moa Apagodu, and Doron Zeilberger, Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle, arXiv:1507.07660 [math.CO], 2015.
J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
Henry Bottomley, Illustration of initial terms
M. Buckley, R. Garner, S. Lack, and R. Street, Skew-monoidal categories and the Catalan simplicial set, arXiv preprint arXiv:1307.0265 [math.CT], 2013.
L. Carlitz, Solution of certain recurrences, SIAM J. Appl. Math., 17 (1969), 251-259.
J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80.
Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
I. Dolinka, J. East, A. Evangelou, D. FitzGerald, and N. Ham, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv preprint arXiv:1507.04838 [math.CO], 2015-2016.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
A. Luzón, D. Merlini, M. A. Morón, and R. Sprugnoli, Complementary Riordan arrays, Discrete Applied Mathematics, 172 (2014) 75-87.
A. Roshan, P. H. Jones and C. D. Greenman, An Exact, Time-Independent Approach to Clone Size Distributions in Normal and Mutated Cells, arXiv preprint arXiv:1311.5769 [q-bio.QM], 2013.
M. János Uray, A family of barely expansive polynomials, Eötvös Loránd University (Budapest, Hungary, 2020).
Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, PDF.
Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, Combinatorics, Probability and Computing, Volume 24, Special Issue 01,January 2015, pp 354-372.
D. Yaqubi, M. Farrokhi D.G., and H. Gahsemian Zoeram, Lattice paths inside a table. I, arXiv:1612.08697 [math.CO], 2016-2017.

Reflected version is in A064189.
Row sums are in A005773.
T(n,n) are Motzkin numbers A001006.
Other columns of T include A002026, A005322, A005323.
Cf. A099323, A097357, A005043, A059738, A027907, A020474, A059738.

a026300 n k = a026300_tabl !! n !! k
a026300_row n = a026300_tabl !! n
a026300_tabl = iterate (\row -> zipWith (+) ([0,0] ++ row) $
                                zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Oct 09 2013

A026300 := proc(n,k)
   add(binomial(n,2*i+n-k)*(binomial(2*i+n-k,i) -binomial(2*i+n-k,i-1)), i=0..floor(k/2));
end proc: # R. J. Mathar, Jun 30 2013

t[n_, k_] := Sum[ Binomial[n, 2i + n - k] (Binomial[2i + n - k, i] - Binomial[2i + n - k, i - 1]), {i, 0, Floor[k/2]}]; Table[ t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jan 03 2011 *)
t[, 0] = 1; t[n, 1] := n; t[n_, k_] /; k>n || k<0 = 0; t[n_, n_] := t[n, n] = t[n-1, n-2]+t[n-1, n-1]; t[n_, k_] := t[n, k] = t[n-1, k-2]+t[n-1, k-1]+t[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 18 2014 *)
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[1/2 - k/2, -k/2, n - k + 2, 4];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Mar 21 2018 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(sum(i=0, k\2, binomial(n, 2*i+n-k)*(binomial(2*i+n-k, i)-binomial(2*i+n-k, i-1))), ", ");); print(););} \\ Michel Marcus, Jul 25 2015

T(n,k) = Sum_{i=0..floor(k/2)} binomial(n, 2i+n-k)*(binomial(2i+n-k, i) - binomial(2i+n-k, i-1)). - Herbert Kociemba, May 27 2004
T(n,k) = A027907(n,k) - A027907(n,k-2), k<=n.
Sum_{k=0..n} (-1)^k*T(n,k) = A099323(n+1). - Philippe Deléham, Mar 19 2007
Sum_{k=0..n} (T(n,k) mod 2) = A097357(n+1). - Philippe Deléham, Apr 28 2007
Sum_{k=0..n} T(n,k)*x^(n-k) = A005043(n), A001006(n), A005773(n+1), A059738(n) for x = -1, 0, 1, 2 respectively. - Philippe Deléham, Nov 28 2009
T(n,k) = binomial(n, k)*hypergeom([1/2 - k/2, -k/2], [n - k + 2], 4). - Peter Luschny, Mar 21 2018
T(n,k) = [t^(n-k)] [x^n] 2/(1 - (2*t + 1)*x + sqrt((1 + x)*(1 - 3*x))). - Peter Luschny, Oct 24 2018
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x^2)*(1 + x + x^2)^n expanded about the point x = 0. - Peter Bala, Feb 26 2023

Corrected and edited by Johannes W. Meijer, Oct 05 2010

A020474 A Motzkin triangle: a(n,k), n >= 2, 2 <= k <= n, = number of complete, strictly subdiagonal staircase functions.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 2, 4, 0, 0, 1, 5, 9, 0, 0, 0, 3, 12, 21, 0, 0, 0, 1, 9, 30, 51, 0, 0, 0, 0, 4, 25, 76, 127, 0, 0, 0, 0, 1, 14, 69, 196, 323, 0, 0, 0, 0, 0, 5, 44, 189, 512, 835, 0, 0, 0, 0, 0, 1, 20, 133, 518, 1353, 2188, 0, 0, 0, 0, 0, 0, 6, 70, 392, 1422, 3610, 5798, 0, 0, 0, 0

Offset: 2

N. J. A. Sloane

T(n,k) = number of Dyck n-paths that start UU, contain no DUDU and no subpath of the form UUPDD with P a nonempty Dyck path and whose terminal descent has length n-k+2. For example, T(5,4)=2 counts UUDUUDUDDD, UUUDDUUDDD (each ending with exactly n-k+2=3 Ds). - David Callan, Sep 25 2006

			Triangle begins:
  1
  0, 1
  0, 1, 2
  0, 0, 2, 4
  0, 0, 1, 5,  9
  0, 0, 0, 3, 12, 21
  0, 0, 0, 1,  9, 30, 51
  0, 0, 0, 0,  4, 25, 76, 127
  0, 0, 0, 0,  1, 14, 69, 196, 323

Reinhard Zumkeller, Rows n = 2..120 of triangle, flattened
M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675.
R. De Castro, A. L. Ramírez and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv preprint arXiv:1310.2449 [cs.DM], 2013.
J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80.
R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
Paul Peart and Wen-jin Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Appl. Math. 98 (2000), 255-263.

Main diagonal is A001006.
Other diagonals include A002026, A005322, A005323, A005324, A005325. Row sums are (essentially) A005043.
The triangle version of A062105 has the same recurrence with different initial conditions. - N. J. A. Sloane, Apr 11 2020

a020474 n k = a020474_tabl !! (n-2) !! (k-2)
a020474_row n = a020474_tabl !! (n-2)
a020474_tabl = map fst $ iterate f ([1], [0,1]) where
   f (us,vs) = (vs, scanl (+) 0 ws) where
     ws = zipWith (+) (us ++ [0]) vs
-- Reinhard Zumkeller, Jan 03 2013

M:=16; T:=Array(0..M,0..M,0);
T[0,0]:=1; T[1,1]:=1;
for i from 1 to M do T[i,0]:=0; od:
for n from 2 to M do for k from 1 to n do
T[n,k]:= T[n,k-1]+T[n-1,k-1]+T[n-2,k-1];
od: od;
rho:=n->[seq(T[n,k],k=0..n)];
for n from 0 to M do lprint(rho(n)); od: # N. J. A. Sloane, Apr 11 2020

a[2,2]=1; a[n_,k_]/;Not[n>2 && 2<=k<=n] := 0; a[n_,k_]/;(n>2 && 2<=k<=n) := a[n,k] = a[n,k-1] + a[n-1,k-1] + a[n-2,k-1]; Table[a[n,k],{n,2,10},{k,2,n}] (* David Callan, Sep 25 2006 *)

T(n,k)=if(n==0&&k==0,1,if(n<=0||k<=0||nRalf Stephan

@cached_function
def T(n, k):
    if k<0 or nPeter Luschny, Jun 23 2015

a(n,k) = a(n,k-1) + a(n-1,k-1) + a(n-2,k-1), n > k >= 2.

More terms from James Sellers, Feb 04 2000

A026109 a(n) = 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) = 3, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n-3), where T is the array defined in A026105.

Original entry on oeis.org

1, 3, 10, 30, 89, 259, 748, 2148, 6150, 17578, 50204, 143364, 409500, 1170300, 3346944, 9579840, 27444681, 78698475, 225887010, 648985414, 1866356437, 5372348487, 15478733108, 44637360700, 128837626255, 372183158061, 1076041247778

Offset: 3

Clark Kimberling

nonn

First differences of A005323. Cf. A026124.

G.f.: z(1-z)M^4, with M the g.f. of the Motzkin numbers (A001006).

Conjecture: (n+5)*a(n) +5*(-n-3)*a(n-1) +4*n*a(n-2) +8*n*a(n-3) +(-5*n+19)*a(n-4) +3*(-n+5)*a(n-5)=0. - R. J. Mathar, Jun 23 2013

A106489 Triangle read by rows: T(n,k) is the number of short bushes with n edges and having the leftmost leaf at height k (a short bush is an ordered tree with no nodes of outdegree 1).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 9, 5, 1, 21, 12, 3, 51, 30, 9, 1, 127, 76, 25, 4, 323, 196, 69, 14, 1, 835, 512, 189, 44, 5, 2188, 1353, 518, 133, 20, 1, 5798, 3610, 1422, 392, 70, 6, 15511, 9713, 3915, 1140, 230, 27, 1, 41835, 26324, 10813, 3288, 726, 104, 7, 113634, 71799, 29964

Offset: 2

Emeric Deutsch, May 29 2005

Basically, the mirror image of A020474. Row n has floor(n/2) terms (first row is row 2). Row sums yield the Riordan numbers (A005043). Column 1 yields the Motzkin numbers (A001006); column 2 yields A002026; column 3 yields A005322; column 4 yields A005323; column 4 yields A005324; column 5 yields A005325; column 6 yields A005326.

T(n,k) is the number of Riordan paths (Motzkin paths with no flatsteps on the x-axis) with k returns to the x-axis. For example, T(6,2) = 5 counts UDUFFD, UDUUDD, UFDUFD, UFFDUD, UUDDUD where U = (1,1) is an upstep, F = (1,0) is a flatstep, and D = (1,-1) is a downstep. - David Callan, Dec 12 2021

			Column 1 yields the Motzkin numbers: indeed, if from each short bush, having leftmost leaf at height 1, we drop the leftmost edge, then we obtain the so-called bushes, known to be counted by the Motzkin numbers.
Triangle begins:
   1;
   1;
   2,  1;
   4,  2;
   9,  5,  1;
  21, 12,  3;
  51, 30,  9,  1.

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
Sen-Peng Eu, Shu-Chung Liu, and Yeong-Nan Yeh, Taylor expansions for Catalan and Motzkin numbers, Advances in Applied Mathematics 29, Issue 3 (2002), 345-357 (see p. 352).

Cf. A020474, A005043, A001006, A002026, A005322, A005323, A005324, A005325, A005326, A064189.

S:=1/2/(z+z^2)*(1+z-sqrt(1-2*z-3*z^2)): G:=simplify(t*z^2*S/(1-z*S-t*z^2*S)): Gserz:=simplify(series(G,z=0,19)): for n from 2 to 17 do P[n]:=sort(coeff(Gserz,z^n)) od: for n from 2 to 17 do seq(coeff(P[n],t^k),k=1..floor(n/2)) od; # yields sequence in triangular form

(* To generate the sequence *)
CoefficientList[CoefficientList[Series[(1-t-2xt^2-Sqrt[1-2t-3t^2])/(2t^2(1-x+xt+x^2t^2)), {t,0,10}], t], x] // Flatten
(* To generate the triangle *)
CoefficientList[Series[(1-t-2xt^2-Sqrt[1-2t-3t^2])/(2t^2(1-x+xt+x^2t^2)), {t, 0, 10}], {t, x}] // MatrixForm
Table[If[n < 2 k, 0, GegenbauerC[n-2k,-n+k-1,-1/2](k+1)/(n-k+1)], {n,0,10}, {k,0,5}] // MatrixForm
(* Emanuele Munarini, Feb 10 2018 *)

G.f.: tz^2*S/(1 - zS - tz^2*S), where S = S(z) = (1 + z - sqrt(1 - 2z - 3z^2))/(2z(1+z)) is the g.f. of the short bushes (the Riordan numbers; A005043).
a(n,k) = T(n-k+1, n-2*k)*(k+1)/(n-k+1), for n >= 2k, where T(n,k) = A027907(n,k) are the trinomial coefficients. - Emanuele Munarini, Feb 10 2018
The rows are the antidiagonals of the Motzkin triangle A064189. - Peter Luschny, Feb 01 2025

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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A026300 Motzkin triangle, T, read by rows; T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1).

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A020474 A Motzkin triangle: a(n,k), n >= 2, 2 <= k <= n, = number of complete, strictly subdiagonal staircase functions.

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A026109 a(n) = 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) = 3, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n-3), where T is the array defined in A026105.

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A106489 Triangle read by rows: T(n,k) is the number of short bushes with n edges and having the leftmost leaf at height k (a short bush is an ordered tree with no nodes of outdegree 1).

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A123261 Multiplicative encoding of Motzkin triangle (A026300).

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