A005554 -id:A005554 - cp's OEIS frontend

A081741 Numbers k that divide A005554(k) (the sum of consecutive Motzkin numbers).

1, 2, 3, 6, 9, 18, 27, 54, 81, 102, 126, 162, 189, 243, 369, 486, 567, 729, 1071, 1134, 1323, 1458, 1462, 1539, 2187, 2538, 2889, 3213, 4374, 4617, 6561, 7749, 8262, 8667, 13122, 14238, 15498, 19683, 24786, 27702, 28994, 32319, 34263, 39366, 40838

Offset: 1

Benoit Cloitre, Apr 07 2003

nonn

All the powers of 3 (A000244) are in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..165

Cf. A000244, A001006, A005554.

s = {1, 2}; k1 = 1; k2 = 2; Do[k3 = (2*n*k2 + (3*n - 9)*k1)/(n + 1); If[Divisible[k3, n], AppendTo[s, n]]; k1 = k2; k2 = k3, {n, 3, 42000}]; s (* Amiram Eldar, Jun 28 2022 *)

A014138 Partial sums of (Catalan numbers starting 1, 2, 5, ...).

Original entry on oeis.org

0, 1, 3, 8, 22, 64, 196, 625, 2055, 6917, 23713, 82499, 290511, 1033411, 3707851, 13402696, 48760366, 178405156, 656043856, 2423307046, 8987427466, 33453694486, 124936258126, 467995871776, 1757900019100

Offset: 0

N. J. A. Sloane

Number of paths starting from the root in all ordered trees with n+1 edges (a path is a nonempty tree with no vertices of outdegree greater than 1). Example: a(2)=8 because the five trees with three edges have altogether 1+0+2+2+3=8 paths hanging from the roots. - Emeric Deutsch, Oct 20 2002
a(n) is the sum of the mean maximal pyramid size over all Dyck (n+1)-paths. Also, a(n) = sum of the mean maximal sawtooth size over all Dyck (n+1)-paths. A pyramid (resp. sawtooth) in a Dyck path is a subpath of the form U^k D^k (resp. (UD)^k) with k>=1 and k is its size. For example, the maximal pyramids in the Dyck path uUUDD|UD|UDdUUDD are indicated by uppercase letters (and separated by a vertical bar). Their sizes are 2,1,1,2 left to right and the mean maximal pyramid size of the path is 6/4 = 3/2. Also, the mean maximal sawtooth size of this path is (1+2+1)/3 = 4/3. - David Callan, Jun 07 2006
p^2 divides a(p-1) for prime p of form p=6k+1 (A002476(k)). - Alexander Adamchuk, Jul 03 2006
p^2 divides a(p^2-1) for prime p>3. p^2 divides a(p^3-1) for prime p=7,13,19,... prime p in the form p=6k+1. - Alexander Adamchuk, Jul 03 2006
Row sums of triangle A137614. - Gary W. Adamson, Jan 30 2008
Equals INVERTi transform of A095930: (1, 4, 15, 57, 220, 859, ...). - Gary W. Adamson, May 15 2009
a(n) < A000108(n+1), therefore A176137(n) <= 1. - Reinhard Zumkeller, Apr 10 2010
a(n) is also the sum of the numbers in Catalan's triangle (A009766) from row 0 to row n. - Patrick Labarque, Jul 27 2010
Equals the Catalan sequence starting (1, 1, 2, ...) convolved with A014137 starting (1, 2, 4, 9, ...). - Gary W. Adamson, May 20 2013
p divides a((p-3)/2) for primes {11,23,47,59,...} = A068231 primes congruent to 11 mod 12. - Alexander Adamchuk, Dec 27 2013
a(n) is the number of parking functions of size n avoiding the patterns 132, 213, and 231. - Lara Pudwell, Apr 10 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000(terms 0 to 200 computed by T. D. Noe)
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)
Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.
Kevin Topley, Computationally Efficient Bounds for the Sum of Catalan Numbers, arXiv:1601.04223 [math.CO], 2016.

Cf. A000108, A002476, A005554, A068231, A095930, A137614, A155587.

a014138 n = a014138_list !! n
a014138_list = scanl1 (+) a000108_list  -- Reinhard Zumkeller, Mar 01 2013

a:=n->sum((binomial(2*j,j)/(j+1)),j=1..n): seq(a(n), n=0..24); # Zerinvary Lajos, Dec 01 2006
# Second program:
A014138 := series(exp(2*x)*(BesselI(0, 2*x)/2 - BesselI(1, 2*x)) + exp(x)*(3/2*int(BesselI(0, 2*x)*exp(x), x) - 1/2), x = 0, 26):
seq(n!*coeff(A014138, x, n), n = 0 .. 24); # Mélika Tebni, Aug 31 2024

Table[Sum[(2k)!/k!/(k+1)!,{k,1,n}],{n,1,70}] (* Alexander Adamchuk, Jul 03 2006 *)
Join[{0},Accumulate[CatalanNumber[Range[30]]]] (* Harvey P. Dale, Jan 25 2013 *)
CoefficientList[Series[(1 - 2 x - (1 - 4 x)^(1/2))/(2 x (1 - x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 21 2015 *)
a[0] := 0; a[n_] := Sum[CatalanNumber[k], {k, 1, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jan 14 2017 *)

Vec((1-2*x-(1-4*x)^(1/2))/(2*x*(1-x))) \\ Charles R Greathouse IV, Feb 11 2011

from _future_ import division
A014138_list, b, s = [0], 1, 0
for n in range(1,10**2):
    s += b
    A014138_list.append(s)
    b = b*(4*n+2)//(n+2) # Chai Wah Wu, Jan 28 2016

a(n) = A014137(n)-1.
G.f.: (1-2*x-sqrt(1-4x))/(2x(1-x)) = (C(x)-1)/(1-x) where C(x) is the generating function for the Catalan numbers. - Rocio Blanco, Apr 02 2007
a(n) = Sum_{k=1..n} A000108(k). - Alexander Adamchuk, Jul 03 2006
Binomial transform of A005554: (1, 2, 3, 6, 13, 30, 72, ...). - Gary W. Adamson, Nov 23 2007
D-finite with recurrence: (n+1)*a(n) + (1-5n)*a(n-1) + 2*(2n-1)*a(n-2) = 0. - R. J. Mathar, Dec 14 2011
Equals the Catalan sequence starting (1, 1, 2, ...) convolved with A014137 starting (1, 2, 4, 9, ...). - Gary W. Adamson, May 20 2013
G.f.: 1/x - G(0)/(1-x)/x, where G(k) = 1 - x/(1 - x/(1 - x/(1 - x/G(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Jul 17 2013
G.f.: 1/x - T(0)/(2*x*(1-x)), where T(k) = 2*x*(2*k+1)+ k+2 - 2*x*(k+2)*(2*k+3)/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2013
a(n) ~ 2^(2*n+2)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 10 2013
a(n) = Sum_{i+jA000108. - Yuchun Ji, Jan 10 2019
E.g.f.: exp(2*x)*(BesselI(0, 2*x)/2 - BesselI(1, 2*x)) + exp(x)*(3/2*Integral_{x=-oo..oo} BesselI(0,2*x)*exp(x) dx - 1/2). - Mélika Tebni, Aug 31 2024

Edited by Max Alekseyev, Sep 13 2009 (including adding an initial 0)

Definition edited by N. J. A. Sloane, Oct 03 2009

A333093 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of c(x)^n evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4x))/(2x) is the o.g.f. of the Catalan numbers A000108.

Original entry on oeis.org

1, 2, 8, 41, 232, 1377, 8399, 52138, 327656, 2077934, 13270633, 85226594, 549837391, 3560702069, 23132584742, 150695482041, 984021596136, 6438849555963, 42208999230224, 277144740254566, 1822379123910857, 11998811140766701, 79095365076843134

Offset: 0

Peter Bala, Mar 07 2020

The sequence satisfies the Gauss congruences: a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k.
We conjecture that the sequence satisfies the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. Examples of these congruences are given below.
More generally, for each integer m, we conjecture that the sequence
  {a_m(n) : n >= 0}, defined by setting a_m(n) = the n-th order Taylor polynomial of c(x)^(m*n) evaluated at x = 1, satisfies the same supercongruences. For cases, see A099837 (m = -2), A100219 (m = -1), A000012 (m = 0), A333094 (m = 2), A333095 (m = 3), A333096 (m = 4), A333097 (m = 5).

			n-th order Taylor polynomial of c(x)^n:
  n = 0: c(x)^0 = 1 + O(x)
  n = 1: c(x)^1 = 1 + x + O(x^2)
  n = 2: c(x)^2 = 1 + 2*x + 5*x^2 + O(x^3)
  n = 3: c(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + O(x^4)
  n = 4: c(x)^4 = 1 + 4*x + 14*x^2 + 48*x^3 + 165*x^4 + O(x^5)
Setting x = 1 gives a(0) = 1, a(1) = 1 + 1 = 2, a(2) = 1 + 2 + 5 = 8, a(3) = 1 + 3 + 9 + 28 = 41 and a(4) = 1 + 4 + 14 + 48 + 165 = 232.
The triangle of coefficients of the n-th order Taylor polynomial of c(x)^n, n >= 0, in descending powers of x begins
                                        row sums
  n = 0 |   1                               1
  n = 1 |   1   1                           2
  n = 2 |   5   2    1                      8
  n = 3 |  28   9    3   1                 41
  n = 4 | 165  48   14   4   1            232
   ...
This is a Riordan array belonging to the Hitting time subgroup of the Riordan group. The first column sequence [1, 1, 5, 28, 165, ...] = [x^n] c(x)^n = A025174(n).
Examples of supercongruences:
a(13) - a(1) = 3560702069 - 2 = (3^2)*(13^3)*31*37*157 == 0 ( mod 13^3 ).
a(3*7) - a(3) = 11998811140766701 - 41 = (2^2)*5*(7^4)*32213*7756841 == 0 ( mod 7^3 ).
a(5^2) - a(5) = 22794614296746579502 - 1377 = (5^6)*7*53*6491*605796421 == 0 ( mod 5^6 ).

Peter Bala, Notes on A333093

Cf. A000108, A001006, A001764, A005554, A025174, A333090 through A333097, A372214, A372215.

seq(add(n/(n+k)*binomial(n+2*k-1,k), k = 0..n), n = 1..25);
#alternative program
c:= x -> (1/2)*(1-sqrt(1-4*x))/x:
G := (x,n) -> series(c(x)^n, x, 51):
seq(add(coeff(G(x, n), x, k), k = 0..n), n = 0..25);

Table[SeriesCoefficient[((1 + x)^2 * (1 - Sqrt[(1 - 3*x)/(1 + x)]) / (2*x))^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 28 2020 *)

a(n) = Sum_{k = 0..n} n/(n+k)*binomial(n+2*k-1,k) for n >= 1.
a(n) = [x^n] ( (1 + x)*c(x/(1 + x)) )^n = [x^n] ( (1 + x)*(1 + x*M(x)) )^n, where M(x) = ( 1 - x - sqrt(1 - 2*x - 3*x^2) ) / (2*x^2) is the o.g.f. of the Motzkin numbers A001006.
O.g.f.: ( 1 + x*f'(x)/f(x) )/( 1 - x*f(x) ), where f(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + ... = (1/x)*Revert( x/c(x) ) is the o.g.f. of A001764.
Row sums of the Riordan array ( 1 + x*f'(x)/f(x), x*f(x) ) belonging to the Hitting time subgroup of the Riordan group.
a(n) ~ 3^(3*n + 3/2) / (7 * sqrt(Pi*n) * 2^(2*n+1)). - Vaclav Kotesovec, Mar 28 2020
a(n) = Sum_{k = 0..n} n/(n+2*k)*binomial(n+2*k, k) for n >= 1. - Peter Bala, Apr 20 2024
D-finite with recurrence 2*n*(2*n-1)*(3991*n -21664)*a(n) +(-1329757*n^3 +9119565*n^2 -18270518*n +10657440)*a(n-1) +10*(947050*n^3 -6943257*n^2 +15944396*n -11260008)*a(n-2) +12*(-787878*n^3 +5778161*n^2 -13283386*n +9383340)*a(n-3) +9*(3*n-10)*(3*n-8)*(100503*n -141587)*a(n-4)=0, n>=5. - R. J. Mathar, Nov 22 2024

A059346 Difference array of Catalan numbers A000108 read by antidiagonals.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 2, 3, 5, 3, 4, 6, 9, 14, 6, 9, 13, 19, 28, 42, 15, 21, 30, 43, 62, 90, 132, 36, 51, 72, 102, 145, 207, 297, 429, 91, 127, 178, 250, 352, 497, 704, 1001, 1430, 232, 323, 450, 628, 878, 1230, 1727, 2431, 3432, 4862, 603, 835, 1158, 1608, 2236, 3114

Offset: 0

N. J. A. Sloane, Jan 27 2001

			Array starts:
      1       1       2       5      14      42     132     429
      0       1       3       9      28      90     297    1001
      1       2       6      19      62     207     704    2431
      1       4      13      43     145     497    1727    6071
      3       9      30     102     352    1230    4344   15483
      6      21      72     250     878    3114   11139   40143
     15      51     178     628    2236    8025   29004  105477
     36     127     450    1608    5789   20979   76473  280221
     91     323    1158    4181   15190   55494  203748  751422
    232     835    3023   11009   40304  148254  547674 2031054
    603    2188    7986   29295  107950  399420 1483380 5527750
Triangle starts:
  1;
  0,  1;
  1,  1,  2;
  1,  2,  3,  5;
  3,  4,  6,  9, 14;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
Zhousheng Mei, Suijie Wang, Pattern Avoidance of Generalized Permutations, arXiv:1804.06265 [math.CO], 2018.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024. See p. 36.

Top row is A000108, leading diagonals give A005043, A001006, A005554.
Row sums are A106640.
Cf. A000108, A000245, A026012, A033434, A106534.

T := (n,k) -> (-1)^(n-k)*binomial(2*k,k)*hypergeom([k-n,k+1/2], [k+2], 4)/(k+1): seq(seq(simplify(T(n,k)), k=0..n), n=0..10);
# Peter Luschny, Aug 16 2012, updated May 25 2021

max = 11; t = Table[ Differences[ Table[ CatalanNumber[k], {k, 0, max}], n], {n, 0, max}]; Flatten[ Table[t[[n-k+1, k]], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Nov 15 2011 *)

def T(n, k) :
    if k > n : return 0
    if n == k : return binomial(2*n, n)/(n+1)
    return T(n-1, k) - T(n, k+1)
A059346 = lambda n,k: (-1)^(n-k)*T(n, k)
for n in (0..5): [A059346(n,k) for k in (0..n)] # Peter Luschny, Aug 16 2012

T(n, k) = (-1)^(n-k)*binomial(2*k,k)/(k+1)*hypergeometric([k-n, k+1/2],[k+2], 4). - Peter Luschny, Aug 16 2012

More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001

A080981 A014486-encodings of the trees whose interior zigzag-tree (Stanley's c) is branch-reduced (in the sense defined by Donaghey).

Original entry on oeis.org

0, 2, 10, 12, 44, 50, 52, 178, 180, 204, 210, 216, 228, 716, 722, 728, 740, 818, 820, 844, 866, 868, 872, 914, 920, 932, 2866, 2868, 2892, 2914, 2916, 2920, 2962, 2968, 2980, 3276, 3282, 3288, 3300, 3378, 3380, 3468, 3474, 3480, 3490, 3492, 3504, 3528, 3660

Offset: 0

Antti Karttunen, Mar 02 2003

nonn

Donaghey defines (on page 82 of his paper) the branch-reduced zigzag-trees as those zigzag-trees which do not contain longer than one-edge branches, where a branch is a maximal connected set of edges slanted to the same direction, with no perpendicular edges emanating from its middle. These form the primitive elements of the automorphism A057505/A057506.

Robert Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
Antti Karttunen, Initial terms illustrated in positions 0, 1, 2, 3, 5, 6, 7, 11, 12, 15, 16, 18, 20, 29, 30, 32, ...

a(n) = A014486(A080980(n)). Cf. A080968, A080971. These trees are enumerated by A005554.

a(n) = A014486(A080980(n)).

A102071 Pairwise sums of general ballot numbers (A002026).

Original entry on oeis.org

1, 3, 7, 17, 42, 106, 272, 708, 1865, 4963, 13323, 36037, 98123, 268737, 739833, 2046207, 5682915, 15842505, 44315637, 124348275, 349911204, 987212856, 2791964574, 7913642086, 22477090679, 63964370301, 182353459733, 520735012027, 1489362193002, 4266018891562, 12236183875496, 35142703099692, 101055137177563

Offset: 1

Ralf Stephan, Dec 30 2004

G. C. Greubel, Table of n, a(n) for n = 1..1000
Gennady Eremin, Naturalized bracket row and Motzkin triangle, arXiv:2004.09866 [math.CO], 2020. See Table 2.

First differences of A005554. Partial sums of A026269. 3rd column of A348840.

Cf. A000108, A001006, A002057.

CoefficientList[Series[(4x(1+x))/(1-x+Sqrt[1-2x-3x^2])^2,{x,0,40}],x] (* Harvey P. Dale, Feb 26 2013 *)

a(n):=1/n*sum((binomial(j,n-1-j)+4*binomial(j,n-2-j)+3*binomial(j,n-3-j))*binomial(n,j),j,0,n); /* Vladimir Kruchinin, Mar 08 2016 */

z='z+O('z^66); Vec(4*z*(1+z)/(1-z+sqrt(1-2*z-3*z^2))^2) \\ Joerg Arndt, Mar 08 2016

G.f.: (4*x*(1+x))/(1-x+sqrt(1-2*x-3*x^2))^2.
a(n) = (1/n) * Sum_{j=0..n} ((binomial(j,n-1-j)+4*binomial(j,n-2-j) + 3*binomial(j,n-3-j))*binomial(n,j)). - Vladimir Kruchinin, Mar 08 2016
a(n) ~ 4*3^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 08 2016
a(n) = A001006(n+1) - A001006(n-1). - Gennady Eremin, Sep 23 2021
D-finite with recurrence (n+3)*a(n) + (-3*n-5)*a(n-1) + (-n+3)*a(n-2) + 3*(n-3)*a(n-3) = 0. - R. J. Mathar, Nov 01 2021
From Peter Bala, Feb 02 2024: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*A002057(k).
G.f.: x/(1 + x)*c(x/(1 + x))^4, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

A026269 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0 = s(n), s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also a(n) = T(n,n) and a(n) = Sum{T(k,k-1)}, k = 1,2,...,n, where T is array in A026268.

Original entry on oeis.org

1, 2, 4, 10, 25, 64, 166, 436, 1157, 3098, 8360, 22714, 62086, 170614, 471096, 1306374, 3636708, 10159590, 28473132, 80032638, 225562929, 637301652, 1804751718, 5121677512, 14563448593, 41487279622, 118389089432, 338381552294, 968627180975

Offset: 2

Clark Kimberling

nonn

Convolution of [1,2,3,6,13,..] (A005554) with [1,0,1,2,5,12...] (essentially A002026). - R. J. Mathar, Nov 01 2021

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Gennady Eremin, Arithmetic on Balanced Parentheses: The case of Ordered Motzkin Words, arXiv:1911.01673 [math.CO], 2019. See (4.3) p. 13 (with a different offset).

First differences of A102071.

Drop[CoefficientList[Series[4x^2(1-x^2)/(1-x+Sqrt[1-2x-3x^2])^2, {x,0,30}],x],2] (* Harvey P. Dale, May 05 2011 *)

G.f.: 4z^2(1-z^2)/[1-z+sqrt(1-2z-3z^2)]^2.
D-finite with recurrence (n+2)*a(n) +(-3*n-1)*a(n-1) +(-n+2)*a(n-2) +3*(n-5)*a(n-3)=0. - R. J. Mathar, Jun 10 2013
a(n) ~ 8 * 3^(n-3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014
a(n) = A002026(n-1) - A002026(n-3). - R. J. Mathar, Nov 01 2021

More terms from Ralf Stephan, Dec 30 2004

A091836 A triangle of Motzkin ballot numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 4, 6, 6, 4, 1, 9, 13, 13, 10, 5, 1, 21, 30, 30, 24, 15, 6, 1, 51, 72, 72, 59, 40, 21, 7, 1, 127, 178, 178, 148, 105, 62, 28, 8, 1, 323, 450, 450, 378, 276, 174, 91, 36, 9, 1, 835, 1158, 1158, 980, 730, 480, 273, 128, 45, 10, 1, 2188, 3023, 3023

Offset: 0

Emeric Deutsch, Mar 09 2004

T(n-1,k) is the number of Motzkin paths of length n that have k points on the horizontal axis (besides the first and last point). For example T(1,0)=1 counts the path UD with 2 steps and no intermediate interception with the y=0 axis, and T(1,1)=1 counts the path FF with 2 steps, staying on the y=0 axis. - R. J. Mathar, Jul 23 2017

Riordan matrix A=(g(t),t*g(t)), where g(t)=1+t*M(t)=C(t/(1-t)), where M(t) and C(t) are the g.f. of Motzkin and Catalan numbers. A is a pseudo-involution. - Emanuele Munarini, Jul 03 2024

			Triangle begins:
   1;
   1,  1;
   1,  2,  1;
   2,  3,  3,  1;
   4,  6,  6,  4,  1;
   9, 13, 13, 10,  5,  1;
  21, 30, 30, 24, 15,  6,  1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened)
M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675.
Jean-Luc Baril and Paul Barry, Two kinds of partial Motzkin paths with air pockets, arXiv:2212.12404 [math.CO], 2022.
Richard J. Mathar, Motzkin Islands: a 3-dimensional Embedding of Motzkin Paths, viXra:2009.0152, 2020.
J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.

Mirror image of A034929.
T(n, 0) = A086246(n+1) = A001006(n-1).
T(n, 1) = A005554(n).
Row sums are the Motzkin numbers (A001006).

T[n_, m_] := If[n == m, 1, (-1)^m (m Sum[k (-1)^(n+k) Binomial[n+k-1, n-1] Sum[Binomial[j, -n+m-k+2j] Binomial[n-m, j], {j, 0, n-m}], {k, 1, n-m}])/ (n(n-m))];
Table[T[n, m], {n, 1, 11}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jul 27 2018, after Vladimir Kruchinin *)

T(n,m):=if n=m then 1 else (-1)^m*(m*sum(k*(-1)^(n+k)*binomial(n+k-1,n-1)*sum(binomial(j,-n+m-k+2*j)*binomial(n-m,j),j,0,n-m),k,1,n-m))/(n*(n-m)); /* Vladimir Kruchinin, Aug 20 2012 */

Column k has g.f.: z^k(1+zM)^(k+1).
G.f.: (1+zM)/(1-tz(1+zM)), where M = 1 + zM+ z ^2M^2 is the g.f. of the Motzkin numbers (A001006).
T(n,m) = (m*(Sum_{k=1..n-m} k*(-1)^(n+m+k)*binomial(n+k-1,n-1) * Sum_{j=0..n-m} binomial(j,-n+m-k+2*j)*binomial(n-m,j)))/(n*(n-m)), n>m, T(n,n)=1. - Vladimir Kruchinin, Aug 20 2012
From Emanuele Munarini, Jul 03 2024: (Start)
T(n,k) = Sum_{i=0..n-k} (-1)^(n-k-i)*binomial(n-k-1,n-k-i) * binomial(2*i+k,i+k) * (k+1) / (i+k+1).
T(n,k) = Sum_{i=0..n-k} binomial(n-k-1,n-k-i)*binomial(n-i+1,i)*(k+1)/(n-i+1) for k < n.
T(n,k) = Sum_{i=0..n-k} trinomial(n-k,n-k-i)*binomial(k+1,i)*i/(n-k) for k < n, where trinomial(n,k) = A027907(n,k).
Recurrence: T(n+2,k+2) = T(n+2,k+1) + T(n+1,k+1) - T(n+1,k) - T(n,k). (End)

A378801 G.f. A(x) satisfies A(x) = ( 1 + x/(1 - x*A(x)^(2/3)) )^3.

Original entry on oeis.org

1, 3, 6, 16, 48, 153, 511, 1761, 6219, 22383, 81804, 302766, 1132475, 4274166, 16256685, 62249167, 239772510, 928398831, 3611539758, 14107963848, 55318781982, 217652858539, 859027927911, 3400055112777, 13492710661658, 53673238384560, 213984657134418

Offset: 0

Seiichi Manyama, Dec 09 2024

nonn

Cf. A161634, A365118.

Cf. A005554, A378858.

a(n, r=3, s=1, t=0, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f. A(x) satisfies A(x) = 1 + x * (1 + A(x)^(1/3) + A(x)^(5/3)).
G.f.: A(x) = (1 + x*B(x))^3 where B(x) is the g.f. of A161634.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

A247364 Riordan array (f(x), (f(x)-1)/f(x)) where f(x) = (1 + x - sqrt(1 - 2x - 3x^2))/(2*x).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 4, 3, 1, 1, 9, 9, 6, 4, 1, 1, 21, 21, 15, 8, 5, 1, 1, 51, 51, 36, 22, 10, 6, 1, 1, 127, 127, 91, 54, 30, 12, 7, 1, 1, 323, 323, 232, 142, 75, 39, 14, 8, 1, 1, 835, 835, 603, 370, 205, 99, 49, 16, 9, 1, 1, 2188, 2188, 1585, 983

Offset: 0

Philippe Deléham, Sep 14 2014

This is essentially the reversal of the triangle in A034928, and A204849 with a duplicated first column.

Row sums are A005554(n+1).

			Triangle begins:
1
1, 1
1, 1, 1
2, 2, 1, 1
4, 4, 3, 1, 1
9, 9, 6, 4, 1, 1
21, 21, 15, 8, 5, 1, 1
51, 51, 36, 22, 10, 6, 1, 1
Production matrix begins:
1, 1
0, 0, 1
1, 1, 0, 1
1, 1, 1, 0, 1
1, 1, 1, 1, 0, 1
1, 1, 1, 1, 1, 0, 1
1, 1, 1, 1, 1, 1, 0, 1

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A001006, A005043, A005554, A034928, A086246.

a247364 n k = a247364_tabl !! n !! k
a247364_row n = a247364_tabl !! n
a247364_tabl = [1] : (map reverse a034928_tabf)
-- Reinhard Zumkeller, Sep 20 2014

T(n,0) = A086246(n+1), T(n+1,1) = A001006(n), T(n+2,2) = A005043(n+2).

cp's OEIS Frontend

A081741 Numbers k that divide A005554(k) (the sum of consecutive Motzkin numbers).

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A014138 Partial sums of (Catalan numbers starting 1, 2, 5, ...).

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A333093 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of c(x)^n evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108.

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A059346 Difference array of Catalan numbers A000108 read by antidiagonals.

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A080981 A014486-encodings of the trees whose interior zigzag-tree (Stanley's c) is branch-reduced (in the sense defined by Donaghey).

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A102071 Pairwise sums of general ballot numbers (A002026).

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A026269 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0 = s(n), s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also a(n) = T(n,n) and a(n) = Sum{T(k,k-1)}, k = 1,2,...,n, where T is array in A026268.

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A333093 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of c(x)^n evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4x))/(2x) is the o.g.f. of the Catalan numbers A000108.