A108263 -id:A108263 - cp's OEIS frontend

A086810 Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.

1, 0, 1, 0, 1, 2, 0, 1, 5, 5, 0, 1, 9, 21, 14, 0, 1, 14, 56, 84, 42, 0, 1, 20, 120, 300, 330, 132, 0, 1, 27, 225, 825, 1485, 1287, 429, 0, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 0, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 0, 1, 54, 936, 7644, 34398, 91728

Offset: 0

Philippe Deléham, Aug 05 2003

Mirror image of triangle A133336. - Philippe Deléham, Dec 10 2008
From Tom Copeland, Oct 09 2011: (Start)
With polynomials
P(0,t) = 0
P(1,t) = 1
P(2,t) = t
P(3,t) = t + 2 t^2
P(4,t) = t + 5 t^2 +  5 t^3
P(5,t) = t + 9 t^2 + 21 t^3 + 14 t^4
The o.g.f. A(x,t) = {1+x-sqrt[(1-x)^2-4xt]}/[2(1+t)] (see Drake et al.).
B(x,t)= x-t x^2/(1-x)= x-t(x^2+x^3+x^4+...) is the comp. inverse in x.
Let h(x,t) = 1/(dB/dx) = (1-x)^2/(1+(1+t)*x*(x-2)) = 1/(1-t(2x+3x^2+4x^3+...)), an o.g.f. in x for row polynomials in t of A181289. Then P(n,t) is given by (1/n!)(h(x,t)*d/dx)^n x, evaluated at x=0, A = exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t). These results are a special case of A133437 with u(x,t) = B(x,t), i.e., u_1=1 and (u_n)=-t for n > 1. See A001003 for t = 1. (End)
Let U(x,t) = [A(x,t)-x]/t, then U(x,0) = -dB(x,t)/dt and U satisfies dU/dt = UdU/dx, the inviscid Burgers' equation (Wikipedia), also called the Hopf equation (see Buchstaber et al.). Also U(x,t) = U(A(x,t),0) = U(x+tU,0) since U(x,0) = [x-B(x,t)]/t. - Tom Copeland, Mar 12 2012
Diagonals of A132081 are essentially rows of this sequence. - Tom Copeland, May 08 2012
T(r, s) is the number of [0,r]-covering hierarchies with s segments (see Kreweras). - Michel Marcus, Nov 22 2014
From Yu Hin Au, Dec 07 2019: (Start)
T(n,k) is the number of small Schröder n-paths (lattice paths from (0,0) to (2n,0) using steps U=(1,1), F=(2,0), D=(1,-1) with no F step on the x-axis) that has exactly k U steps.
T(n,k) is the number of Schröder trees (plane rooted tree where each internal node has at least two children) with exactly n+1 leaves and k internal nodes. (End)

			Triangle starts:
  1;
  0,  1;
  0,  1,  2;
  0,  1,  5,  5;
  0,  1,  9, 21, 14;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
Yu Hin (Gary) Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
V. Buchstaber and E. Bunkova,Elliptic formal group laws, integral Hirzebruch genera and Kirchever genera,, arXiv:1010.0944 [math-ph], 2010 (see p. 19).
V. Buchstaber and T. Panov, Toric Topology. Chapter 1: Geometry and Combinatorics of Polytopes,, arXiv:1102.1079 [math.CO], 2011-2012 (see p. 41).
G. Chatel, V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
T. Copeland, Compositional inverse pairs, the Burgers-Hopf equation, and the Stasheff associahedra, 2014.
T. Copeland, Lagrange a la Lah, 2011.
B. Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973, p. 21-22.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
J. Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

Diagonals: A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279, A000108, A002054, A002055, A002056, A007160, A033280, A033281.
The diagonals (except for A000007) are also the diagonals of A033282.
Row sums: A001003 (Schroeder numbers).
Cf. A033282, A084938.
Cf. A001003, A008297, A021009, A132081, A133437, A181289.

Table[Boole[n == 2] + If[# == -1, 0, Binomial[n - 3, #] Binomial[n + # - 1, #]/(# + 1)] &[k - 1], {n, 2, 12}, {k, 0, n - 2}] // Flatten (* after Jean-François Alcover at A033282, or *)
Table[If[n == 0, 1, Binomial[n, k] Binomial[n + k, k - 1]/n], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)

t(n, k) = if (n==0, 1, binomial(n, k)*binomial(n+k, k-1)/n);
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n,k), ", ");); print(););} \\ Michel Marcus, Nov 22 2014

Triangle T(n, k) read by rows; given by [0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
For k>0, T(n, k) = binomial(n-1, k-1)*binomial(n+k, k)/(n+1); T(0, 0) = 1 and T(n, 0) = 0 if n > 0. [corrected by Marko Riedel, May 04 2023]
Sum_{k>=0} T(n, k)*2^k = A107841(n). - Philippe Deléham, May 26 2005
Sum_{k>=0} T(n-k, k) = A005043(n). - Philippe Deléham, May 30 2005
T(n, k) = A108263(n+k, k). - Philippe Deléham, May 30 2005
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Nov 05 2007
Sum_{k=0..n} T(n,k)*5^k*(-2)^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
Sum_{k=0..n} T(n,k)*(-1)^k*3^(n-k) = A154825(n). - Philippe Deléham, Jan 17 2009
Umbrally, P(n,t) = Lah[n-1,-t*a.]/n! = (1/n)*Sum_{k=1..n-1} binomial(n-2,k-1)a_k t^k/k!, where (a.)^k = a_k = (n-1+k)!/(n-1)!, the rising factorial, and Lah(n,t) = n!*Laguerre(n,-1,t) are the Lah polynomials A008297 related to the Laguerre polynomials of order -1. - Tom Copeland, Oct 04 2014
T(n, k) = binomial(n, k)*binomial(n+k, k-1)/n, for k >= 0; T(0, 0) = 1 (see Kreweras, p. 21). - Michel Marcus, Nov 22 2014
P(n,t) = Lah[n-1,-:Dt:]/n! t^(n-1) with (:Dt:)^k = (d/dt)^k t^k = k! Laguerre(k,0,-:tD:) with (:tD:)^j = t^j D^j. The normalized Laguerre polynomials of 0 order are given in A021009. - Tom Copeland, Aug 22 2016

Typo in a(60) corrected by Michael De Vlieger, Nov 21 2019

A100754 Triangle read by rows: T(n, k) = number of hill-free Dyck paths (i.e., no peaks at height 1) of semilength n and having k peaks.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 29, 13, 1, 1, 19, 73, 73, 19, 1, 1, 26, 151, 266, 151, 26, 1, 1, 34, 276, 749, 749, 276, 34, 1, 1, 43, 463, 1781, 2762, 1781, 463, 43, 1, 1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1, 1, 64, 1093, 7253, 21659, 31004, 21659, 7253, 1093, 64, 1

Offset: 2

Emeric Deutsch, Jan 14 2005

Row n has n - 1 terms. Row sums yield the Fine numbers (A000957).
Related to the number of certain sets of non-crossing partitions for the root system A_n (p. 11, Athanasiadis and Savvidou). - Tom Copeland, Oct 19 2014
T(n,k) is the number of permutations pi of [n-1] with k - 1 descents such that s(pi) avoids the patterns 132, 231, and 312, where s is West's stack-sorting map. - Colin Defant, Sep 16 2018
The absolute values of the polynomials at -1 and j (cube root of 1) seem to be given by A126120 and A005043. - F. Chapoton, Nov 16 2021
Don Knuth observes that this sequence also arrises from the enumeration of restricted max-and-min-closed relations, only there it appears as an array read by antidiagonals: see the Knuth "Notes" link and A372068. Knuth also gives a formula expressing the array A372368 in terms of this array. He also reports that there is strong experimental evidence that the n-th term of row m in this array is a polynomial of degree 2*m-2 in n. - N. J. A. Sloane, May 12 2024

			T(4, 2) = 4 because we have UU*DDUU*DD, UU*DUU*DDD, UUU*DDU*DD and UUU*DU*DDD, where U = (1, 1), D = (1,-1) and * indicates the peaks.
Triangle starts:
   1;
   1,  1;
   1,  4,   1;
   1,  8,   8,    1;
   1, 13,  29,   13,    1;
   1, 19,  73,   73,   19,    1;
   1, 26, 151,  266,  151,   26,    1;
   1, 34, 276,  749,  749,  276,   34,   1;
   1, 43, 463, 1781, 2762, 1781,  463,  43,  1;
   1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1;
   ...
As an array (for which the rows of the preceding triangle are the antidiagonals):
   1,  1,    1,     1,      1,      1,       1,        1,        1, ...
   1,  4,    8,    13,     19,     26,      34,       43,       53, ...
   1,  8,   29,    73,    151,    276,     463,      729,     1093, ...
   1, 13,   73,   266,    749,   1781,    3758,     7253,    13061, ...
   1, 19,  151,   749,   2762,   8321,   21659,    50471,   107833, ...
   1, 26,  276,  1781,   8321,  31004,   97754,   271125,   679355, ...
   1, 34,  463,  3758,  21659,  97754,  367285,  1196665,  3478915, ...
   1, 43,  729,  7253,  50471, 271125, 1196665,  4526470, 15118415, ...
   1, 53, 1093, 13061, 107833, 679355, 3478915, 15118415, 57500480, ...
   ...

Alois P. Heinz, Rows n = 2..142, flattened
C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024

Cf. A000957, A108263, A132081.

See also A099594, A272644, A372066, A372067, A372068.

T := (n, k) -> add((j/(n-j))*binomial(n-j, k-j)*binomial(n-j,k), j=0..min(k,n-k)): for n from 2 to 13 do seq(T(n, k), k = 1..n-1) od; # yields the sequence in triangular form

T[n_, k_] := Sum[(j/(n-j))*Binomial[n-j, k-j]*Binomial[n-j, k], {j, 0, Min[k, n-k]}]; Table[T[n, k], {n, 2, 13}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

T(n, k) = Sum_{j=0..min(k, n-k)} (j/(n-j)) * C(n-j, k-j) * C(n-j, k), n >= 2.
G.f.: t*z*r/(1 - t*z*r), where r = r(t, z) is the Narayana function defined by r = z*(1 + r)*(1 + t*r).
From Tom Copeland, Oct 19 2014: (Start)
With offset 0 for A108263 and offset 1 for A132081, row polynomials of this entry P(n, x) = Sum_{i} A108263(n, i)*x^i*(1 + x)^(n - 2*i) = Sum_{i} A132081(n - 2, i)*x^i*(1 + x)^(n - 2*i).
E.g., P(4, x) = 1*x*(1 + x)^(4 - 2*1) + 2*x^2*(1 + x)^(4 - 2*2) = x + 4*x^2 + x^3.
Equivalently, let Q(n, x) be the row polynomials of A108263. Then P(n, x) = (1 + x)^n * Q(n, x/(1 + x)^2).
E.g., P(4, x) = (1 + x)^4 * (x/(1 + x)^2 + 2 * (x/(1 + x)^2)^2).
See Athanasiadis and Savvidou (p. 7). (End)

A033275 Number of diagonal dissections of an n-gon into 3 regions.

Original entry on oeis.org

0, 5, 21, 56, 120, 225, 385, 616, 936, 1365, 1925, 2640, 3536, 4641, 5985, 7600, 9520, 11781, 14421, 17480, 21000, 25025, 29601, 34776, 40600, 47125, 54405, 62496, 71456, 81345, 92225, 104160, 117216, 131461, 146965, 163800, 182040, 201761, 223041, 245960

Offset: 4

N. J. A. Sloane

Number of standard tableaux of shape (n-3,2,2) (n>=5). - Emeric Deutsch, May 13 2004

Number of short bushes with n+1 edges and 3 branch nodes (i.e., nodes with outdegree at least 2). A short bush is an ordered tree with no nodes of outdegree 1. Example: a(5)=5 because the only short bushes with 6 edges and 3 branch nodes are the five full binary trees with 6 edges. Column 3 of A108263. - Emeric Deutsch, May 29 2005

Indranil Ghosh, Table of n, a(n) for n = 4..10000
David Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, Vol. 105, No. 3 (1998), pp. 256-257.
Frank R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., Vol. 204, No. 1-3 (1999), pp. 73-112.
Ronald C. Read, On general dissections of a polygon, Aequat. Math., Vol. 18 (1978), pp. 370-388, Table 1.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

2nd skew subdiagonal of A033282.

Cf. A033276, A108263.

a[4]=0; a[n_]:=Binomial[n+1,2]*Binomial[n-3,2]/3; Table[a[n],{n,4,43}] (* Indranil Ghosh, Feb 20 2017 *)

concat(0, Vec(z^5*(5-4*z+z^2)/(1-z)^5 + O(z^60))) \\ Michel Marcus, Jun 18 2015

a(n) = binomial(n+1, 2)*binomial(n-3, 2)/3 \\ Charles R Greathouse IV, Feb 20 2017

def A033275(n): return (binomial(n+1, 2)*binomial(n-3, 2))//3
print([A033275(n) for n in range(4,50)]) # Peter Luschny, Apr 03 2020

a(n) = binomial(n+1, 2)*binomial(n-3, 2)/3.
G.f.: z^5*(5-4*z+z^2)/(1-z)^5. - Emeric Deutsch, May 29 2005
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=5} 1/a(n) = 43/150.
Sum_{n>=5} (-1)^(n+1)/a(n) = 16*log(2)/5 - 154/75. (End)
E.g.f.: x*(exp(x)*(12 - 6*x + x^3) - 6*(2 + x))/12. - Stefano Spezia, Feb 21 2024

A132081 Triangle (read by rows) with row sums = Motzkin sums (also called Riordan numbers) (A005043): T(n,s) = (1/n)C(n,s)(C(n-s,s+1) - C(n-s-2,s-1)).

Original entry on oeis.org

1, 1, 2, 1, 5, 1, 9, 5, 1, 14, 21, 1, 20, 56, 14, 1, 27, 120, 84, 1, 35, 225, 300, 42, 1, 44, 385, 825, 330, 1, 54, 616, 1925, 1485, 132, 1, 65, 936, 4004, 5005, 1287, 1, 77, 1365, 7644, 14014, 7007, 429

Offset: 3

Frank R. Bernhart (farb45(AT)gmail.com), Oct 30 2007

Whereas A005043 counts certain trees, or noncrossed partitions, this subdivides the counts according to the number of leaves, or the lattice rank. Analogous to the Narayana triangle (A001263), where rows sum to the Catalan numbers.
Diagonals of A132081 are rows of A033282. - Tom Copeland, May 08 2012
Related to the number of certain non-crossing partitions for the root system A_n. Cf. p. 12, Athanasiadis and Savvidou. See also A108263 and A100754. - Tom Copeland, Oct 19 2014

			A005043(6) = 15 = 1+9+5 since NC (noncrossed, planar) partitions of 6-point cycle without singletons have 1,9,5 items with 1,2,3 blocks.
Triangle begins:
  1;
  1,   2;
  1,   5;
  1,   9,   5;
  1,  14,  21;
  1,  20,  56,  14;
  1,  27, 120,  84;
  1,  35, 225, 300,  42;
  1,  44, 385, 825, 330;
  ...

C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.

Row sums are A007404.

Cf. A001263, A005043, A033282, A100754, A108263, A108767, A132081.

/* triangle excluding 0 */ [[Binomial(n,k)*Binomial(n-2-k,k)/(k+1): k in [0..n-3]]: n in [3..15]]; // Vincenzo Librandi, Oct 19 2014

Map[Most, Table[(1/n) Binomial[n, s] (Binomial[n - s, s + 1] - Binomial[n - s - 2, s - 1]), {n, 3, 14}, {s, 0, n}] /. k_ /; k <= 0 -> Nothing] // Flatten (* Michael De Vlieger, Jan 09 2016 *)

a(n,k) = binomial(n,k)*binomial(n-2-k,k)/(k+1). - David Callan, Jul 22 2008
From Peter Bala, Oct 22 2008: (Start)
O.g.f.: (1 + x - sqrt(1 - 2*x + x^2*(1 - 4*a)))/(2*x*(1 + a*x)) = 1 + a*x^2 + a*x^3 + (a + 2*a^2)*x^4 + (a + 5*a^2)*x^5 + (a + 9*a^2 + 5*a^3)*x^6 + ... . [corrected by Jason Yuen, Sep 22 2024]
Define a functional I on formal power series of the form f(x) = 1 + a*x + b*x^2 + ... by the following iterative process. Define inductively f^(1)(x) = f(x) and f^(n+1)(x) = f(x*f^(n)(x)) for n >= 1. Then set I(f(x)) = lim n -> infinity f^(n)(x) in the x-adic topology on the ring of formal power series; the operator I may also be defined by I(f(x)) := 1/x*series reversion of x/f(x).
Let now f(x) = 1 + a*x^2 + a*x^3 + a*x^4 + ... . Then the o.g.f. for this table is I(f(x)) = 1 + a*x^2 + a*x^3 + (a + 2*a^2)*x^4 + (a + 5*a^2)*x^5 + (a + 9*a^2 + 5*a^3)*x^6 + ... . Cf. A001263 and A108767. (End)

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar

Name corrected by Emeric Deutsch, Dec 20 2014

A350248 Triangle read by rows: T(n,k) is the number of noncrossing partitions of an n-set into k blocks of size 3 or more, n >= 0, 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 3, 0, 1, 7, 0, 1, 12, 0, 1, 18, 12, 0, 1, 25, 45, 0, 1, 33, 110, 0, 1, 42, 220, 55, 0, 1, 52, 390, 286, 0, 1, 63, 637, 910, 0, 1, 75, 980, 2275, 273, 0, 1, 88, 1440, 4900, 1820, 0, 1, 102, 2040, 9520, 7140, 0, 1, 117, 2805, 17136, 21420, 1428

Offset: 0

Andrew Howroyd and Janaka Rodrigo, Dec 21 2021

			Triangle begins:
  1;
  0;
  0;
  0, 1;
  0, 1;
  0, 1;
  0, 1,   3;
  0, 1,   7;
  0, 1,  12;
  0, 1,  18,   12;
  0, 1,  25,   45;
  0, 1,  33,  110;
  0, 1,  42,  220,   55;
  0, 1,  52,  390,  286;
  0, 1,  63,  637,  910;
  0, 1,  75,  980, 2275,  273;
  0, 1,  88, 1440, 4900, 1820;
  0, 1, 102, 2040, 9520, 7140;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1750 (rows 0..100)

Columns k=2..5 are A055998, A350116, A350286, A350303.
Row sums are A114997.
Cf. A001263 (blocks of any size), A108263 (blocks of size 2 or more).

T(n)={my(p=1+O(x^3)); for(i=1, n\3, p=1+y*(x*p)^3/(1-x*p)); [Vecrev(t)| t<-Vec(p + O(x*x^n))]}
{my(A=T(12)); for(i=1, #A, print(A[i]))}

T(n,k) = if(n==0 || k>n\3, k==0, binomial(n+1, n-k+1) * binomial(n-2*k-1, k-1) / (n+1)) \\ Andrew Howroyd, Dec 31 2021

G.f.: A(x,y) satisfies A(x,y) = 1 + y*(x*A(x,y))^3/(1 - x*A(x,y)).

T(n,k) = binomial(n+1, n-k+1) * binomial(n-2*k-1, k-1) / (n+1) for n > 0.

A033276 Number of diagonal dissections of an n-gon into 4 regions.

Original entry on oeis.org

0, 14, 84, 300, 825, 1925, 4004, 7644, 13650, 23100, 37400, 58344, 88179, 129675, 186200, 261800, 361284, 490314, 655500, 864500, 1126125, 1450449, 1848924, 2334500, 2921750, 3627000, 4468464, 5466384, 6643175, 8023575, 9634800, 11506704, 13671944

Offset: 5

N. J. A. Sloane

Number of standard tableaux of shape (n-4,2,2,2) (n>=6). - Emeric Deutsch, May 20 2004

Number of short bushes with n+2 edges and 4 branch nodes (i.e. nodes with outdegree at least 2). A short bush is an ordered tree with no nodes of outdegree 1. Example: a(6)=14 because the only short bushes with 8 edges and 4 branch nodes are the fourteen full binary trees with 8 edges. Column 4 of A108263. - Emeric Deutsch, May 29 2005

Vincenzo Librandi, Table of n, a(n) for n = 5..1000
David Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, Vol. 105, No. 3 (1998), 256-257.
Frank R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., Vol. 204, No. 1-3 (1999), 73-112.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978), 370-388, Table 1.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A033275, A108263.

[(Binomial(n+2,3)*Binomial(n-3,3))/4: n in [5..50]]; // Vincenzo Librandi, Mar 15 2014

Table[(Binomial[n+2,3]Binomial[n-3,3])/4,{n,5,40}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,14,84,300,825,1925,4004},40] (* Harvey P. Dale, Mar 13 2014 *)
CoefficientList[Series[x (14 - 14 x + 6 x^2 - x^3)/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *)

a(n) = binomial(n+2, 3)*binomial(n-3, 3)/4.
G.f.: x^6*(14-14x+6x^2-x^3)/(1-x)^7. - Emeric Deutsch, May 29 2005
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=6} 1/a(n) = 109/1225.
Sum_{n>=6} (-1)^n/a(n) = 192*log(2)/35 - 4582/1225. (End)

More terms from Vincenzo Librandi, Mar 15 2014

A033278 Number of diagonal dissections of an n-gon into 6 regions.

Original entry on oeis.org

0, 132, 1287, 7007, 28028, 91728, 259896, 659736, 1534896, 3325608, 6789783, 13180167, 24496472, 43835792, 75869640, 127481640, 208606320, 333316620, 521215695, 799197399, 1203649524, 1783184480, 2601993680, 3743934480, 5317472160, 7461614160, 10352989647

Offset: 7

N. J. A. Sloane

nonn

Number of standard tableaux of shape (n-6,2,2,2,2,2) (n>=8). - Emeric Deutsch, May 20 2004

Number of short bushes with n+4 edges and 6 branch nodes (i. e. nodes with outdegree at least 2; a short bush is an ordered tree with no nodes of outdegree 1). Example: a(8)=132 because the only short bushes with 12 edges and 6 branch nodes are the one-hundred-thirty-two full binary trees with 12 edges. Column 6 of A108263. - Emeric Deutsch, May 29 2005

T. D. Noe, Table of n, a(n) for n = 7..1000
D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.

Cf. A108263.

vector(40, n, n+=6; binomial(n+4, 5)*binomial(n-3, 5)/6) \\ Michel Marcus, Jun 18 2015

a(n) = binomial(n+4, 5)*binomial(n-3, 5)/6.

G.f.: z^8(132-165z+110z^2-44z^3+10z^4-z^5)/(1-z)^11. - Emeric Deutsch, May 29 2005

A033279 Number of diagonal dissections of an n-gon into 7 regions.

Original entry on oeis.org

0, 429, 5005, 32032, 148512, 556920, 1790712, 5116320, 13302432, 32008977, 72177105, 153977824, 313112800, 610569960, 1147334760, 2086063200, 3682355040, 6329047725, 10617908301, 17424259776, 28021470400, 44233892560, 68638798800, 104830165440, 157759842240

Offset: 8

N. J. A. Sloane

nonn

Number of standard tableaux of shape (n-7,2,2,2,2,2,2) (n>=9). - Emeric Deutsch, May 21 2004

Number of short bushes with n+5 edges and 7 branch nodes (i.e. nodes with outdegree at least 2; a short bush is an ordered tree with no nodes of outdegree 1). Example: a(9)=429 because the only short bushes with 14 edges and 7 branch nodes are the four-hundred-twenty-nine full binary trees with 14 edges. Column 7 of A108263. - Emeric Deutsch, May 29 2005

T. D. Noe, Table of n, a(n) for n = 8..1000
D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.

Cf. A108263.

Table[(Binomial[n+5,6]Binomial[n-3,6])/7,{n,8,40}] (* Harvey P. Dale, May 27 2013 *)

vector(40, n, n+=7; binomial(n+5, 6)*binomial(n-3, 6)/7) \\ Michel Marcus, Jun 18 2015

a(n) = binomial(n+5, 6)*binomial(n-3, 6)/7.

G.f.: z^9(429-572z+429z^2-208z^3+65z^4-12z^5+z^6)/(1-z)^13. - Emeric Deutsch, May 29 2005

A033277 Number of diagonal dissections of an n-gon into 5 regions.

Original entry on oeis.org

0, 42, 330, 1485, 5005, 14014, 34398, 76440, 157080, 302940, 554268, 969969, 1633905, 2662660, 4214980, 6503112, 9806280, 14486550, 21007350, 29954925, 42063021, 58241106, 79606450, 107520400, 143629200, 189909720, 248720472, 322858305, 415621185, 530877480

Offset: 6

N. J. A. Sloane

nonn

Number of standard tableaux of shape (n-5,2,2,2,2) (n>=7). - Emeric Deutsch, May 20 2004

Number of short bushes with n+3 edges and 5 branch nodes (i.e. nodes with outdegree at least 2; a short bush is an ordered tree with no nodes of outdegree 1). Example: a(7)=42 because the only short bushes with 10 edges and 5 branch nodes are the fortytwo full binary trees with 10 edges. Column 5 of A108263. - Emeric Deutsch, May 29 2005

D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.

Cf. A108263.

vector(40, n, n+=5; binomial(n+3, 4)*binomial(n-3, 4)/5) \\ Michel Marcus, Jun 18 2015

a(n) = binomial(n+3, 4)*binomial(n-3, 4)/5.

G.f.: z^7(42-48z+27z^2-8z^3+z^4)/(1-z)^9. - Emeric Deutsch, May 29 2005

cp's OEIS Frontend

A086810 Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.

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A100754 Triangle read by rows: T(n, k) = number of hill-free Dyck paths (i.e., no peaks at height 1) of semilength n and having k peaks.

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A033275 Number of diagonal dissections of an n-gon into 3 regions.

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A132081 Triangle (read by rows) with row sums = Motzkin sums (also called Riordan numbers) (A005043): T(n,s) = (1/n)*C(n,s)*(C(n-s,s+1) - C(n-s-2,s-1)).

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A350248 Triangle read by rows: T(n,k) is the number of noncrossing partitions of an n-set into k blocks of size 3 or more, n >= 0, 0 <= k <= floor(n/3).

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A033276 Number of diagonal dissections of an n-gon into 4 regions.

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A033278 Number of diagonal dissections of an n-gon into 6 regions.

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A132081 Triangle (read by rows) with row sums = Motzkin sums (also called Riordan numbers) (A005043): T(n,s) = (1/n)C(n,s)(C(n-s,s+1) - C(n-s-2,s-1)).