A067323 -id:A067323 - cp's OEIS frontend

A067324 Third column of triangle A067323.

2, 7, 23, 76, 255, 869, 3003, 10504, 37128, 132430, 476102, 1723528, 6277505, 22988385, 84592275, 312636240, 1159979700, 4319218530, 16134883410, 60452176200, 227110782990, 855361970034, 3228982640478

Offset: 0

Wolfdieter Lang, Feb 05 2002

Also third diagonal of triangle A028364.

Cf. A000245 (second column).

a(n)= A067323(n+2, 2)= C(n+3)-(C(n+2)+C(n+1)), C(n) := A000108(n) (Catalan).

G.f.: (c(x)^3)*(1+c(x)), with c(x) g.f. of A000108 (Catalan).

A067325 Fourth column of triangle A067323.

Original entry on oeis.org

5, 19, 66, 227, 785, 2739, 9646, 34268, 122706, 442510, 1605956, 5861481, 21502585, 79243395, 293246550, 1089264360, 4059928950, 15179606010, 56917649820, 213982542150, 806429435994, 3046017513198

Offset: 0

Wolfdieter Lang, Feb 05 2002

Also fourth diagonal of triangle A028364.

Cf. A067324 (third column).

a(n)= A067323(n+3, 3)= C(n+4)-(C(n+3)+C(n+2)+2*C(n+1)), C(n) := A000108(n) (Catalan).

G.f.: (c(x)^3)*(2+2*c(x)+c(x)^2), with c(x) g.f. of A000108 (Catalan).

A067326 Fifth column of triangle A067323.

Original entry on oeis.org

14, 56, 202, 715, 2529, 8986, 32123, 115556, 418200, 1521976, 5567551, 20462525, 75528895, 279874350, 1040790135, 3883140600, 14531382060, 54529456320, 205146226200, 773608833894, 2923686178098, 11071970477876

Offset: 0

Wolfdieter Lang, Feb 05 2002

Also fifth diagonal of triangle A028364.

Cf. A067325 (fourth column).

a(n)= A067323(n+4, 4)= C(n+5)-sum(C(k)*C(n+4-k), k=0..3), C(n) := A000108(n) (Catalan).

G.f.: (c(x)^3)*(5+5*c(x)+3*c(x)^2+c(x)^3), with c(x) g.f. of A000108 (Catalan).

A116871 Sixth column of triangle A067323.

Original entry on oeis.org

42, 174, 645, 2333, 8398, 30275, 109550, 398180, 1453908, 5332407, 19639521, 72616727, 269473750, 1003347975, 3747412770, 14036374680, 52714429260, 198459284400, 748867149234, 2831788492218, 10729442739596

Offset: 0

Wolfdieter Lang, Mar 24 2006

Also sixth diagonal sequence of triangle A028364.

Cf. A067326 (fifth column of A067323).

a(n)= A067323(n+5,5), n>=0.
a(n)= A028364(5+n,n) = sum(C(k)*C(5+n-k),k=0..n), n>=0, with the Catalan numbers C(n):=A000108(n).
G.f.: (c(x)^3)sum(C(4, k)*c(x)^k, k=0..4), with C(n, m) := (m+1)*binomial(2*n-m, n-m)/(n+1) (Catalan convolutions A033184).

A000245 a(n) = 3(2n)!/((n+2)!*(n-1)!).

Original entry on oeis.org

0, 1, 3, 9, 28, 90, 297, 1001, 3432, 11934, 41990, 149226, 534888, 1931540, 7020405, 25662825, 94287120, 347993910, 1289624490, 4796857230, 17902146600, 67016296620, 251577050010, 946844533674, 3572042254128, 13505406670700, 51166197843852, 194214400834356

Offset: 0

N. J. A. Sloane

This sequence represents the expected saturation of a binary search tree (or BST) on n nodes times the number of binary search trees on n nodes, or alternatively, the sum of the saturation of all binary search trees on n nodes. - Marko Riedel, Jan 24 2002
1->12, 2->123, 3->1234 etc. starting with 1, gives A007001: 1, 12, 12123, 12123121231234... summing the digits gives this sequence. - Miklos Kristof, Nov 05 2002
a(n-1) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 2 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003
With offset 1, number of permutations beginning with 12 and avoiding 32-1.
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=1. - Herbert Kociemba, May 24 2004
a(n) is the number of Dyck (n+1)-paths that start with UU. For example, a(2)=3 counts UUUDDD, UUDUDD, UUDDUD. - David Callan, Dec 08 2004
a(n) is the number of Dyck (n+2)-paths that start with UUDU. For example, a(2)=3 counts UUDUDDUD, UUDUDUDD, UUDUUDDD. - David Scambler, Feb 14 2011
Hankel transform is (0,-1,-1,0,1,1,0,-1,-1,0,...). Hankel transform of a(n+1) is (1,0,-1,-1,0,1,1,0,-1,-1,0,...). - Paul Barry, Feb 08 2008
Starting with offset 1 = row sums of triangle A154558. - Gary W. Adamson, Jan 11 2009
Starting with offset 1 equals INVERT transform of A014137, partial sums of the Catalan numbers: (1, 2, 4, 9, 23, ...). - Gary W. Adamson, May 15 2009
With offset 1, a(n) is the binomial transform of the shortened Motzkin numbers: 1, 2, 4, 9, 21, 51, 127, 323, ... (A001006). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Sep 07 2009
The Catalan sequence convolved with its shifted variant, e.g. a(5) = 90 = (1, 1, 2, 5, 14) dot (42, 14, 5, 2, 1) = (42 + 14 + 10 + 10 + 14 ) = 90. - Gary W. Adamson, Nov 22 2011
a(n+2) = A214292(2*n+3,n). - Reinhard Zumkeller, Jul 12 2012
With offset 3, a(n) is the number of permutations on {1,2,...,n} that are 123-avoiding, i.e., do not contain a three term monotone subsequence, for which the first ascent is at positions (3,4); see Connolly link. There it is shown in general that the k-th Catalan Convolution is the number of 123-avoiding permutations for which the first ascent is at (k, k+1). (For n=k, the first ascent is defined to be at positions (k,k+1) if the permutation is the decreasing permutation with no ascents.) - Anant Godbole, Jan 17 2014
With offset 3, a(n)=number of permutations on {1,2,...,n} that are 123-avoiding and for which the integer n is in the 3rd spot; see Connolly link. For example, there are 297 123-avoiding permutations on n=9 at which the element 9 is in the third spot. - Anant Godbole, Jan 17 2014
With offset 1, a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that bounce off y = x to the right exactly once but do not cross y = x vertically. Details can be found in Section 4.4 in Pan and Remmel's link. - Ran Pan, Feb 01 2016
The total number of returns (downsteps which end on the line y=0) within the set of all Dyck paths from (0,0) to (2n,0). - Cheyne Homberger, Sep 05 2016
a(n) is the number of intervals of the form [s,w] that are distributive (equivalently, modular) lattices in the weak order on S_n, for a fixed simple reflection s. - Bridget Tenner, Jan 16 2020
a(n+2) is the number of coalescent histories for a pair (G,S) where G is a gene tree with 3-pseudocaterpillar shape and n leaves, S is a species tree with caterpillar shape and n leaves, and G and S have identical leaf labeling. - Noah A Rosenberg, Jun 15 2022
a(n) is the number of parking functions of size n avoiding the patterns 132, 213, and 312. - Lara Pudwell, Apr 10 2023
a(n) is the number of parking functions of size n avoiding the patterns 123 and 213. - Lara Pudwell, Apr 10 2023

Pierre de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 11, coefficients of P_3(z).
Ki Hang Kim, Douglas G. Rogers and Fred W. Roush, Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577-594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013)
C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
Marco Abrate, Stefano Barbero, Umberto Cerruti and Nadir Murru, Colored compositions, Invert operator and elegant compositions with the "black tie", Discrete Math. 335 (2014), 1-7. MR3248794
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
F. S. Al-Kharousi, A. Umar, and M. M. Zubairu, On injective partial Catalan monoids, arXiv:2501.00285 [math.GR], 2024. See p. 2.
Egor Alimpiev and Noah A Rosenberg, Enumeration of coalescent histories for caterpillar species trees and p-pseudocaterpillar gene trees, arXiv:2103.13464 [qbio.PE], 2021; Adv. Appl. Math. 131 (2021), 102265.
Jean-Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
Jean-Luc Baril and Sergey Kirgizov, The pure descent statistic on permutations, Preprint, 2016.
Jean-Luc Baril and Helmut Prodinger, Enumeration of partial Lukasiewicz paths, arXiv:2205.01383 [math.CO], 2022.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
David Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], Nov 25 2002.
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.
S. Connolly, Z. Gabor and A. Godbole, The location of the first ascent in a 123-avoiding permutation, arXiv:1401.2691 [math.CO], 2014.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 196).
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 31.
Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro and Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8.
Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
Filippo Disanto and Thomas Wiehe, Some instances of a sub-permutation problem on pattern avoiding permutations, arXiv preprint arXiv:1210.6908 [math.CO], 2012-2014.
Filippo Disanto and Thomas Wiehe, On the sub-permutations of pattern avoiding permutations, Discrete Math., 337 (2014), 127-141.
Manuel Flores, Yuta Kimura and Baptiste Rognerud, Combinatorics of quasi-hereditary structures, arXiv:2004.04726 [math.RT], 2020.
Sela Fried and Toufik Mansour, Counting r X s rectangles in (Catalan) words, arXiv:2405.06962 [math.CO], 2024. See p. 11.
Alice L.L. Gao, Sergey Kitaev, and Philip B. Zhang, On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
Guo-Niu Han, Enumeration of Standard Puzzles [broken link]
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
Sergey Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003). See also arXiv:0205215 [math.CO], 2002.
Sergey Kitaev and Toufik Mansour, Simultaneous avoidance of generalized patterns, arXiv:math/0205182 [math.CO], 2002.
C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146. [Annotated scanned copy]
D. Nečas, Elementary models of low-pressure plasma polymerisation into nanofibrous mats, Phys. Scr. (2025) Vol. 100, 055601. See p. 10.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
A. Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]
A. Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math. 14 (1956), 405ff.
J.-B. Priez and A. Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv:1411.4161 [math.CO], 2014-2015.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
J. Riordan, Letter to N. J. A. Sloane, Nov 10 1970
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.
S. J. Tedford, Combinatorial interpretations of convolutions of the Catalan numbers, Integers 11 (2011) #A3.
Bridget E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
Qi Wang, Tau-tilting finite simply connected algebras, arXiv:1910.01937 [math.RT], 2019.

First differences of Catalan numbers A000108.
T(n, n+3) for n=0, 1, 2, ..., array T as in A047072.
Cf. A099364.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A002057, A000344, A003517, A000588, A003518, A003519, A001392, A001622.
Cf. A154558, A014137.
Column k=1 of A067323.

Concatenation([0],List([1..30],n->3*Factorial(2*n)/(Factorial(n+2)*Factorial(n-1)))); # Muniru A Asiru, Aug 09 2018

[0] cat [3*Factorial(2*n)/(Factorial(n+2)*Factorial(n-1)): n in [1..30]]; // Vincenzo Librandi, Feb 15 2016

A000245 := n -> 3*binomial(2*n, n-1)/(n+2);
seq(A000245(n), n=0..27);

Table[3(2n)!/((n+2)!(n-1)!),{n,0,30}] (* or *) Table[3*Binomial[2n,n-1]/(n+2),{n,0,30}] (* or *) Differences[CatalanNumber[Range[0,31]]] (* Harvey P. Dale, Jul 13 2011 *)

PARI
```
a(n)=if(n<1,0,3*(2*n)!/(n+2)!/(n-1)!)
```

[catalan_number(i+1) - catalan_number(i) for i in range(28)] # Zerinvary Lajos, May 17 2009

def A000245_list(n) :
    D = [0]*(n+1); D[1] = 1
    b = False; h = 1; R = []
    for i in range(2*n-1) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1; R.append(D[2])
        else :
            for k in range(1,h, 1) : D[k] += D[k+1]
        b = not b
    return R
A000245_list(29) # Peter Luschny, Jun 03 2012

a(n) = A000108(n+1) - A000108(n).
G.f.: x*(c(x))^3 = (-1+(1-x)*c(x))/x, c(x) = g.f. for Catalan numbers. Also a(n) = 3*n*Catalan(n)/(n+2). - Wolfdieter Lang
For n > 1, a(n) = 3a(n-1) + Sum[a(k)*a(n-k-2), k=1,...,n-3]. - John W. Layman, Dec 13 2002; proved by Michael Somos, Jul 05 2003
G.f. is A(x) = C(x)*(1-x)/x-1/x = x(1+x*C(x)^2)*C(x)^2 where C(x) is g.f. for Catalan numbers, A000108.
G.f. satisfies x^2*A(x)^2 + (3*x-1)*A(x) + x = 0.
Series reversion of g.f. A(x) is -A(-x). - Michael Somos, Jan 21 2004
a(n+1) = Sum_{i+j+k=n} C(i)C(j)C(k) with i, j, k >= 0 and where C(k) denotes the k-th Catalan number. - Benoit Cloitre, Nov 09 2003
An inverse Chebyshev transform of x^2. - Paul Barry, Oct 13 2004
The sequence is 0, 0, 1, 0, 3, 0, 9, 0, ... with zeros restored. Second binomial transform of (-1)^n*A005322(n). The g.f. is transformed to x^2 under the Chebyshev transformation A(x)->(1/(1+x^2))A(x/(1+x^2)). For a sequence b(n), this corresponds to taking Sum_{k=0..floor(n/2)} C(n-k, k)(-1)^k*b(n-2k), or Sum_{k=0..n} C((n+k)/2, k)*b(k)*(-1)^((n-k)/2)*(1+(-1)^(n-k))/2. - Paul Barry, Oct 13 2004
G.f.: (c(x^2)*(1-x^2)-1)/x^2, c(x) the g.f. of A000108; a(n) = Sum_{k=0..n} (k+1)*C(n, (n-k)/2)*(-1)^k*(C(2,k)-2*C(1,k)+C(0, k))*(1+(-1)^(n-k))/(n+k+2). - Paul Barry, Oct 13 2004
a(n) = Sum_{k=0..n} binomial(n,k)*2^(n-k)*(-1)^(k+1)*binomial(k, floor((k-1)/2)). - Paul Barry, Feb 16 2006
E.g.f.: exp(2*x)*(Bessel_I(1,2x) - Bessel_I(2,2*x)). - Paul Barry, Jun 04 2007
a(n) = (1/Pi)*Integral_{x=0..4} x^n*(x-1)*sqrt(x*(4-x))/(2*x). - Paul Barry, Feb 08 2008
D-finite with recurrence: For n > 1, a(n+1) = 2*(2n+1)*(n+1)*a(n)/((n+3)*n). - Sean A. Irvine, Dec 09 2009
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j] = Catalan(j-i), (i<=j), and A[i,j] = 0, otherwise. Then, for n >= 2, a(n-1) = (-1)^(n-2)*coeff(charpoly(A,x),x^2). - Milan Janjic, Jul 08 2010
a(n) = sum of top row terms of M^(n-1), M = an infinite square production matrix as follows:
  2, 1, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, 1, ...
  ...
- Gary W. Adamson, Jul 14 2011
E.g.f.: exp(2*x)*(BesselI(2,2*x)) = Q(0) - 1 where Q(k) = 1 - 2*x/(k + 1 - 3*((k+1)^2)/((k^2) + 8*k + 9 - (k+2)*((k+3)^2)*(2*k+3)/((k+3)*(2*k+3) - 3*(k+1)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Dec 05 2011
a(n) = -binomial(2*n,n)/(n+1)*hypergeom([-1,n+1/2],[n+2],4). - Peter Luschny, Aug 15 2012
a(n) = Sum_{i=0..n-1} C(i)*C(n-i), where C(i) denotes the i-th Catalan number. - Dmitry Kruchinin, Mar 02 2013
a(n) = binomial(2*n-1, n) - binomial(2*n-1, n-3). - Johannes W. Meijer, Jul 31 2013
a(n) ~ 3*4^n/(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Feb 26 2016
a(n) = ((-1)^n/(n+1))*Sum_{i=0..n-1} (-1)^(i+1)*(n+1-i)*binomial(2*n+2,i), n>=0. - Taras Goy, Aug 09 2018
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=1} 1/a(n) = 14*Pi/(27*sqrt(3)) + 5/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 164*log(phi)/(75*sqrt(5)) + 7/25, where phi is the golden ratio (A001622). (End)
a(n) = 3*Sum_{k = 0..n-2} (-1)^k * binomial(2*n-k-1, n+1)*binomial(n+1, k)/(k + 1) for n >= 2. - Peter Bala, Sep 02 2024
a(n) = (3*n/(n+2))*A000108(n). - Taras Goy, Dec 20 2024

I changed the description and added an initial 0, to make this coincide with the first differences of the Catalan numbers A000108. Some of the other lines will need to be changed as a result. - N. J. A. Sloane, Oct 31 2003

A028364 Triangle T(n,m) = Sum_{k=0..m} Catalan(n-k)*Catalan(k).

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 5, 7, 9, 14, 14, 19, 23, 28, 42, 42, 56, 66, 76, 90, 132, 132, 174, 202, 227, 255, 297, 429, 429, 561, 645, 715, 785, 869, 1001, 1430, 1430, 1859, 2123, 2333, 2529, 2739, 3003, 3432, 4862, 4862, 6292, 7150, 7810, 8398, 8986, 9646, 10504, 11934, 16796

Offset: 0

Wouter Meeussen

There are several versions of a Catalan triangle: see A009766, A008315, A028364.
The subtriangle [1], [2, 3], [5, 7, 9], ..., namely T(N,M-1), for N >= 1, M=1..N, appears as one-point function in the totally asymmetric exclusion process for the parameters alpha=1=beta. See the Derrida et al. and Liggett references given under A067323, where these triangle entries are called T_{N,N+M-1} for the given alpha and beta values. See the row reversed triangle A067323.
Consider a Dyck path as a path with steps N=(0,1) and E=(1,0) from (0,0) to (n,n) that stays weakly above y=x. T(n,m) is the number of Dyck paths of semilength n+1 where the (m+1)st north step is followed by an east step. - Lara Pudwell, Apr 12 2023

			Triangle begins
   1;
   1,  2;
   2,  3,  5;
   5,  7,  9, 14;
  14, 19, 23, 28, 42;

Alois P. Heinz, Rows n = 0..140, flattened
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
G. Chatel and V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.
A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741.

Cf. A000108 (column 0 and main diagonal), A001700 (row sums), A065097 (T(2*n-1, n-1)), A201205 (central terms).

Cf. A067323, A009766, A039598, A039599, A028377, A028378, A028376.

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      expand(b(n-1, j)*`if`(i>n, x, 1)), j=1..i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b((n+1)$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 28 2015

t[n_, k_] = Sum[CatalanNumber[n-j]*CatalanNumber[j], {j, 0, k}]; Flatten[Table[t[n, k], {n, 0, 8}, {k, 0, n}]] (* Jean-François Alcover, Jul 22 2011 *)

T(n,k) = Sum_{j>=0} A039598(k,j)*A039599(n-k,j). - Philippe Deléham, Feb 18 2004
Sum_{k>=0} T(n,k) = A001700(n). T(n,k) = A067323(n,n-k), n >= k >= 0, otherwise 0. - Philippe Deléham, May 26 2005
G.f. for column sequences m >= 0: (-(c(m,x)-1)/x+c(m,x)*c(x))/x^m with the g.f. c(x) of A000108 (Catalan) and c(m,x):=sum(C(k)*x^k,k=0..m) with C(n):=A000108(n). - Wolfdieter Lang, Mar 24 2006
G.f. for column sequences m >= 0 (without leading zeros): c(x)*Sum_{k=0..m} C(m,k)*c(x)^k with the g.f. c(x) of A000108 (Catalan) and C(n,m) is the Catalan triangle A033184(n,m). - Wolfdieter Lang, Mar 24 2006
T(n,n) = T(n,k) + T(n,n-1-k) = A000108(n+1), n > 0, k = 0..floor((n+1)/2). - Yuchun Ji, Jan 09 2019
G.f. for triangle: Sum_{n>=0, m>=0} T(n, m)*x^n*y^m = (c(x)-c(xy))/(x(1-y)c(x)) with the g.f. c(x) of A000108 (Catalan). - Lara Pudwell, Apr 12 2023

A067304 Generalized Catalan triangle A067298 with row reversion.

Original entry on oeis.org

1, 2, 1, 9, 5, 4, 64, 36, 32, 28, 584, 328, 300, 284, 256, 6144, 3440, 3184, 3072, 2960, 2704, 70576, 39408, 36704, 35680, 34896, 33872, 31168, 859520, 478912, 447744, 436928, 429760, 422592, 411776, 380608, 10909440, 6068480, 5687872, 5563200, 5487488, 5421952, 5346240, 5221568, 4840960

Offset: 0

Wolfdieter Lang, Feb 05 2002

Identity for each row n >= 1: T(n, m) + T(n, n-m+1) = A067297(n+1) (convolution of generalized Catalan numbers) for every m = 1..floor((n+1)/2). E.g., T(2*k+1, k+1) = A067297(2*(k+1))/2.

			Triangle begins:
    1;
    2,   1;
    9,   5,   4;
   64,  36,  32,  28;
  584, 328, 300, 284, 256;
  ...
n=3: T(3, 0) = 64 = 36+28 = 32+32.

The columns give for m=0..4: A067297 (diagonals of A067298), A067305, A067306, A067307, A067308.
Cf. A067302 (row sums), A067323 (corresponding triangle for ordinary Catalan numbers).
Cf. A000108, A067298, A067329.

T(n, m) = A067298(n, n-m), n >= m >= 0, otherwise 0.

G.f. for column m >= 1 (without leading zeros): (2^(2*ceiling(m/2))*p(m, y)*(y^3)/(1+y)^4, where y = y(x) = c(4*x), with c(x) = g.f. of A000108 (Catalan) and the row polynomials p(n, y) = Sum_{k=0..n} A067329(n, k)*y^k, n >= 1. For m = 0: ((y*(3+y))^2)/(1+y)^4 with y = y(x) = c(4*x) (see A067297).

More terms from Jinyuan Wang, Apr 20 2025

A201205 Bisection of half-convolution of Catalan sequence A000108; even part.

Original entry on oeis.org

1, 3, 23, 227, 2529, 30275, 380162, 4939443, 65844845, 895451117, 12374186318, 173257703723, 2452607696798, 35042725663002, 504697422982484, 7319313029400467, 106793147620036005, 1566546633240722681, 23089471526179716182, 341774295456352388245

Offset: 0

Wolfdieter Lang, Jan 02 2012

For the definition of the half-convolution of a sequence with itself see a comment to A201204.

The odd part of this bisection is found under A065097.

Alois P. Heinz, Table of n, a(n) for n = 0..800

Cf. A000108, A001246, A028364, A067323, A201204, A065097.

a:= proc(n) option remember; `if`(n<2, 1+2*n,
      (2*n*(256*n^5-544*n^4+256*n^3+75*n^2-69*n+12)*a(n-1)
       -(8*(4*n-5))*(4*n-3)*(8*n^2+n-1)*(2*n-3)^2*a(n-2))/
      ((2*n+1)*n*(8*n^2-15*n+6)*(n+1)^2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 28 2015

Table[(CatalanNumber[2 n + 1] + CatalanNumber[n]^2)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)

a(n) = sum(Catalan(k)*Catalan(2*n-k),k=0..n), n>=0, with Catalan(n)=A000108(n).
O.g.f: Ge(x)=(catao(x)+cata2(x))/2 with catao(x):= sum(Catalan(2*k+1)*x^k,k=0..infty) = (cata(sqrt(x)) - cata(-sqrt(x)))/(2*x), with the o.g.f. cata(x) of A000108, and cata2(x):=sum(Catalan(n)^2,n=0..infty) given in A001246 as (-1 + hypergeom( [-1/2,-1/2],[1],16*x))/(4*x).
a(n) = A028364(2n,n) = A067323(2n,n). - Alois P. Heinz, Nov 28 2015
a(n) = (A000108(2*n+1) + A000108(n)^2)/2. - Vladimir Reshetnikov, Oct 03 2016

Cross-reference corrected by Robert Israel, Jun 06 2014

A116872 Subtriangle of generalized Catalan triangle CM(1,2) = A116880.

Original entry on oeis.org

1, 3, 7, 13, 29, 41, 67, 147, 195, 247, 381, 829, 1069, 1277, 1545, 2307, 4995, 6339, 7379, 8451, 9975, 14589, 31485, 39549, 45373, 50733, 56829, 66057, 95235, 205059, 255747, 290691, 320707, 351187, 388099

Offset: 1

Wolfdieter Lang, Mar 24 2006

This triangle a(n,m) appears for the unnormalized one-point function T(n,n+m-1) in the totally asymmetric exclusion process (see A067323 for the references) for the (unphysical) values alpha=1, beta=2.

W. Lang: First 10 rows.

Row sums give A116879.

a(n,m)=A116880(n,m-1), n>=m>=1.

G.f. for column m>=1: (x^m)*(-(C2(m) + ((2^2)/x^(m-1))*(c(m-1,2*x)-1)/(2*x)) + 2*(C2(m-1) + (2/x^(m-1))*c(m-2,2*x))*c(2*x))/(1+x) where C2(n):=A064062(n), c(m,x):=sum(C(k)*x^k,k=0..m) with C(k):=A000108(k) (Catalan numbers) and c(x) is the g.f. of A000108.

A116869 Sixth column of triangle A028364.

Original entry on oeis.org

132, 297, 869, 2739, 8986, 30275, 104006, 362738, 1280592, 4566642, 16423581, 59497075, 216901454, 795117015, 2929034130, 10837075830, 40253156520, 150045410430, 561097823490, 2104382963214, 7913603390196

Offset: 0

Wolfdieter Lang, Mar 24 2006

Also sixth diagonal of triangle A067323.

Cf. A067296 (fifth column).

a(n)= A028364(n+5,5) = sum(C(k)*C(n+5-k),k=0..5), with the Catalan numbers C(n):=A000108(n).

G.f.: (c5(x)*c(x)-(c5(x)-1)/x)/x^5, with c5(x) := sum(C(k)*x^k, k=0..5) and c(x) is the g.f. for the Catalan numbers A000108.

cp's OEIS Frontend

A067324 Third column of triangle A067323.

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A067325 Fourth column of triangle A067323.

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A067326 Fifth column of triangle A067323.

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A116871 Sixth column of triangle A067323.

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A000245 a(n) = 3*(2*n)!/((n+2)!*(n-1)!).

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A028364 Triangle T(n,m) = Sum_{k=0..m} Catalan(n-k)*Catalan(k).

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A067304 Generalized Catalan triangle A067298 with row reversion.

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A201205 Bisection of half-convolution of Catalan sequence A000108; even part.

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A000245 a(n) = 3(2n)!/((n+2)!*(n-1)!).