A004321 -id:A004321 - cp's OEIS frontend

A002694 Binomial coefficients C(2n, n-2).

1, 6, 28, 120, 495, 2002, 8008, 31824, 125970, 497420, 1961256, 7726160, 30421755, 119759850, 471435600, 1855967520, 7307872110, 28781143380, 113380261800, 446775310800, 1761039350070, 6943526580276, 27385657281648, 108043253365600

Offset: 2

N. J. A. Sloane

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=2. Example: For n=3 there are 6 paths EEENNN, EENENN, EENNEN, EENNNE, ENEENN and NEEENN. - Herbert Kociemba, May 23 2004
Number of dissections of a convex (n+3)-gon by noncrossing diagonals into several regions, exactly n-2 of which are triangular. Example: a(3)=6 because the convex hexagon ABCDEF is dissected by any of the diagonals AC, BD, CE, DF, EA, FB into regions containing exactly 1 triangle. - Emeric Deutsch, May 31 2004
Number of UUU's (triple rises), where U=(1,1), in all Dyck paths of semilength n+1. Example: a(3)=6 because we have UD(UUU)DDD, (UUU)DDDUD, (UUU)DUDDD, (UUU)DDUDD and (U[UU)U]DDDD, the triple rises being shown between parentheses. - Emeric Deutsch, Jun 03 2004
Inverse binomial transform of A026389. - Ross La Haye, Mar 05 2005
Sum of the jump-lengths of all full binary trees with n internal nodes. In the preorder traversal of a full binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given full binary tree is called the jump-length. - Emeric Deutsch, Jan 18 2007
a(n) = number of convex polyominoes (A005436) of perimeter 2n+4 that are directed but not parallelogram polyominoes, because the directed convex polyominoes are counted by the central binomial coefficient binomial(2n,n) and the subset of parallelogram polyominoes is counted by the Catalan number C(n+1) = binomial(2n+2,n+1)/(n+2) and a(n) = binomial(2n,n) - C(n+1). - David Callan, Nov 29 2007
a(n) = number of DUU's in all Dyck paths of semilength n+1. Example: a(3)=6 because we have UU(DUU)DDD, U(DUU)UDDD, U(DUU)DUDD, UDU(DUU)DD, U(DUU)DDUD, UUD(DUU)DD, the DUU's being shown between parentheses and no other Dyck path of semilength 4 contains a DUU. - David Callan, Jul 25 2008
C(2n,n-m) is the number of Dyck-type walks such that their graphs have one marked edge passed 2m times and the other edges are passed 2 times counting "there and back" directions. - Oleksiy Khorunzhiy, Jan 09 2015
Number of paths in the half-plane x >= 0, from (0,0) to (2n,4), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=3, we have the 6 paths: UUUUUD, UUUUDU, UUUDUU, UUDUUU, UDUUUU, DUUUUU, DUUUUU. - José Luis Ramírez Ramírez, Apr 19 2015

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 517.
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 = 2..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Henry Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311
Milan Janjic, Two Enumerative Functions
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.
O. Khorunzhiy, On high moments and the spectral norm of large dilute Wigner random matrices, arXiv:1107.5724 [math-ph], 2014.
O. Khorunzhiy, On high moments and the spectral norm of large dilute Wigner random matrices, Zh. Mat. Fiz. Anal. Geom. 10 (1) (2014), pp. 64-125.
W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Article 12.3.3, 2012. - From _N. J. A. Sloane_, Sep 16 2012
V. Pilaud and J. Rué, Analytic combinatorics of chord and hyperchord diagrams with k crossings, arXiv:1307.6440 [math.CO], 2013; Adv. Appl. Math. 57 (2014) 60-100.
Mark Shattuck, Enumeration of non-crossing partitions according to subwords with repeated letters, arXiv:2303.06300 [math.CO], 2023.
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).
T. Tao and Van Vu, Random matrices: localization of the eigenvalues and the necessity of four moments, arXiv:1005.2901 [math.PR], 2010-2011; Acta Math. Vietnam 36 (2) (2011) 431-449.
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.

Cf. A006659.
Diagonal 5 of triangle A100257.
Cf. A001622, A009766.
Cf. binomial(k*n, n-k): A000027 (k=1), this sequence (k=2), A004321 (k=3), A004334 (k=4), A004347 (k=5), A004361 (k=6), A004375 (k=7), A004389 (k=8), A281580 (k=9).
Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A003516 (m = 5), A002696 (m = 6), A030053 - A030056, A004310 - A004318.

List([2..30], n-> Binomial(2*n,n-2)); # G. C. Greubel, Mar 21 2019

a002694 n = a007318' (2 * n) (n - 2)  -- Reinhard Zumkeller, Jun 18 2012

[Binomial(2*n, n-2): n in [2..30]]; // Vincenzo Librandi, Apr 20 2015

a:=n->sum(binomial(n,j-1)*binomial(n,j+1),j=1..n): seq(a(n), n=2..25); # Zerinvary Lajos, Nov 26 2006

CoefficientList[ Series[ 16/(((Sqrt[1 - 4 x] + 1)^4)*Sqrt[1 - 4 x]), {x, 0, 23}], x] (* Robert G. Wilson v, Aug 08 2011 *)
Table[Binomial[2n,n-2],{n,2,30}] (* Harvey P. Dale, Jun 12 2014 *)

{a(n) = binomial(2*n,n-2)}; \\ G. C. Greubel, Mar 21 2019

[binomial(2*n,n-2) for n in (2..30)] # G. C. Greubel, Mar 21 2019

a(n) = A067310(n, 1) as this is number of ways of arranging n chords on a circle (handshakes between 2n people across a table) with exactly 1 simple intersection. - Henry Bottomley, Oct 07 2002
E.g.f.: exp(2*x) * BesselI(2, 2*x). - Vladeta Jovovic, Aug 21 2003
G.f.: (1-sqrt(1-4*z))^4/(16*z^2*sqrt(1-4*z)). - Emeric Deutsch, Jan 28 2004
a(n) = Sum_{k=0..n} C(n, k)*C(n, k+2). - Paul Barry, Sep 20 2004
D-finite with recurrence: -(n-2)*(n+2)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 04 2012
G.f.: z^2*C(z)^4/(1-2*z*C(z)), where C(z) is the g.f. of Catalan numbers. - José Luis Ramírez Ramírez, Apr 19 2015
a(n) = Sum_{k=1..n} binomial(2*n-k,n-k-1). - Vladimir Kruchinin, Oct 22 2016
G.f.: x^2* 2F1(5/2,3;5;4*x). - R. J. Mathar, Jan 27 2020
From Amiram Eldar, May 16 2022: (Start)
Sum_{n>=2} 1/a(n) = 23/6 - 13*Pi/(9*sqrt(3)).
Sum_{n>=2} (-1)^n/a(n) = 106*log(phi)/(5*sqrt(5)) - 37/10, where phi is the golden ratio (A001622). (End)
From Peter Bala, Oct 13 2024: (Start)
a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * (x^2 - 4*x + 2)/sqrt(x*(4 - x)).
G.f. x^2 * B(x) * C(x)^4, where B(x) = 1/sqrt(1 - 4*x) is the g.f. of the central binomial coefficients A000984 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

A023822 Sum of exponents in prime-power factorization of C(3n,n-3).

Original entry on oeis.org

0, 3, 3, 6, 5, 7, 7, 10, 9, 10, 9, 12, 9, 12, 12, 14, 13, 15, 12, 19, 15, 17, 17, 18, 18, 20, 19, 20, 16, 21, 20, 23, 20, 21, 20, 23, 18, 23, 21, 25, 26, 27, 27, 30, 27, 28, 28, 31, 27, 31, 27, 33, 31, 33, 31, 32, 31, 33, 31, 34, 30, 36, 34, 34, 31, 33, 31, 37, 32, 35, 34, 37, 36, 37, 37, 39, 35, 37

Offset: 3

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 3..10000

Cf. A001222, A004321, A023819, A023820, A023822, A023823, A023824, A023825.

Join[{0}, Table[Total[FactorInteger[Binomial[3 n, n - 3]][[All, 2]]], {n, 4, 80}]] (* Ivan Neretin, Nov 02 2017 *)
a[n_] := PrimeOmega[Binomial[3*n, n-3]]; Array[a, 100, 3] (* Amiram Eldar, Jun 12 2025 *)

a(n) = bigomega(binomial(3*n, n-3)); \\ Amiram Eldar, Jun 12 2025

From Amiram Eldar, Jun 12 2025: (Start)
a(n) = A001222(A004321(n)).
a(n) = A023821(n) - A001222(2*n+3) + A001222(n-2). (End)

Offset corrected to 3 by Ivan Neretin, Nov 02 2017

A004334 Binomial coefficient C(4n,n-4).

Original entry on oeis.org

1, 20, 276, 3276, 35960, 376992, 3838380, 38320568, 377348994, 3679075400, 35607051480, 342700125300, 3284214703056, 31368725759168, 298824321028320, 2840671544105280, 26958221130508525, 255485622301674660

Offset: 4

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).

Cf. binomial(k*n, n-k): A000027 (k=1), A002694 (k=2), A004321 (k=3), this sequence (k=4), A004347 (k=5), A004361 (k=6), A004375 (k=7), A004389 (k=8), A281580 (k=9).

List([4..30], n-> Binomial(4*n,n-4)); # G. C. Greubel, Mar 21 2019

[Binomial(4*n, n-4): n in [4..30]]; // Vincenzo Librandi, Feb 01 2017

Table[Binomial[4n, n-4], {n,4,30}] (* Vincenzo Librandi, Feb 01 2017 *)

a(n)=binomial(4*n,n-4) \\ Charles R Greathouse IV, Feb 01 2017

[binomial(4*n,n-4) for n in (4..30)] # G. C. Greubel, Mar 21 2019

From Ilya Gutkovskiy, Jan 31 2017: (Start)
E.g.f.: (1/24)*x^4*3F3(17/4,9/2,19/4; 17/3,6,19/3; 256*x/27).
a(n) ~ 2^(8*n+1/2)/(sqrt(Pi*n)*3^(3*n+9/2)). (End)
D-finite with recurrence -3*(3*n+2)*(n-4)*(3*n+4)*(n+1)*a(n) +8*n*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Mar 19 2025

A120983 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k vertices of outdegree 3 (n >= 0, k >= 0).

Original entry on oeis.org

1, 3, 12, 54, 1, 261, 12, 1323, 105, 6939, 810, 3, 37341, 5859, 63, 205011, 40824, 840, 1143801, 277830, 9072, 12, 6466230, 1861380, 86670, 360, 36960300, 12335895, 764478, 6435, 213243435, 81120204, 6377778, 89100, 55, 1240219269, 530408736

Offset: 0

Emeric Deutsch, Jul 21 2006

A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child.

			T(3,1)=1 because we have (Q,L,M,R), where Q denotes the root and L (M,R) denotes a left (middle, right) child of Q.
Triangle starts:
     1;
     3;
    12;
    54,   1;
   261,  12;
  1323, 105;
  6939, 810, 3;

Cf. A001764, A107264, A120429, A120981, A120982, A004321.

T:=(n,k)->(1/(n+1))*binomial(n+1,k)*sum(3^j*binomial(n+1-k,j)*binomial(j,n-3*k-j),j=0..n+1-k): for n from 0 to 14 do seq(T(n,k),k=0..floor(n/3)) od; # yields sequence in triangular form

T(n,k) = (1/(n+1))*binomial(n+1,k)*Sum_{j=0..n+1-k} 3^j*binomial(n+1-k, j)*binomial(j, n-3k-j).
G.f.: G = G(t,z) satisfies G = 1 + 3zG + 3z^2*G^2 + tz^3*G^3.
Row n has 1+floor(n/3) terms.
Row sums yield A001764.
T(n,0) = A107264(n).
Sum_{k>=1} k*T(n,k) = binomial(3n, n-3) = A004321(n).

A281580 a(n) = binomial(9*n, n-9).

Original entry on oeis.org

1, 90, 4851, 204156, 7413705, 244222650, 7511839335, 219683466288, 6183023199255, 168899639028120, 4505395859893071, 117891537949758600, 3036500678480436531, 77190387796530738576, 1940723247304668029175, 48339506285032758609456, 1194448077521704400002650

Offset: 9

Vincenzo Librandi, Feb 02 2017

Row 9*n, column n-9 of A007318. - Felix Fröhlich, Feb 05 2017

Cf. sequences with formula binomial(k*n, n-k): A002694 (k=2), A004321 (k=3), A004334 (k=4), A004347 (k=5), A004361 (k=6), A004375 (k=7), A004389 (k=8), this sequence (k=9).

Magma
```
[Binomial(9*n, n-9): n in [9..30]];
```
Mathematica
```
Table[Binomial[9 n, n - 9], {n, 9, 25}]
```

a(n) = binomial(9*n, n-9) \\ Felix Fröhlich, Feb 05 2017

A354622 Irregular triangle read by rows: Refined 3-Narayana triangle. Coefficients of partition polynomials of A134264, a refined Narayana triangle enumerating noncrossing partitions, with all h_k other than h_0, h_3, h_6, ..., h_(3n), ... replaced by zero.

Original entry on oeis.org

1, 1, 3, 1, 9, 12, 1, 12, 6, 66, 55, 1, 15, 15, 105, 105, 455, 273, 1, 18, 18, 9, 153, 306, 51, 816, 1224, 3060, 1428, 1, 21, 21, 21, 210, 420, 210, 210, 1330, 3990, 1330, 5985, 11970, 20349, 7752, 1, 24, 24, 24, 12, 276, 552, 552, 276, 276, 2024, 6072, 3036, 6072, 506, 10626, 42504, 21252, 42504, 106260, 134596, 43263

Offset: 1

Tom Copeland, Jul 08 2022

A set of partition polynomials with these coefficients and the polynomials of A338135 can be generated by substitution of the refined Narayana, or noncrossing partition, polynomials N_n[h_1,...,h_n] of A134264 (h_0=1) into themselves--once for A338135 and twice for this entry--or by setting the indeterminates h_n of A134264 to zero except for h_0, h_3, h_6, ..., h_(3n), ... with h_0 = 1 and ultimately re-indexing. This is equivalent to recursive use of the Lagrange inversion formula on f(x) = x / h(x) = x / (1 + h_1 x + h_2 x^2 + ...) since its compositional inverse is f^{(-1)}(x) = x + N_1(h_1) x + N_2(h_1,h_2) x^2 + .... The equivalence of the two methods of generation--the substitution and the zeroing out--follows from the general theorems stated by Peter Bala in his presentation of formulas for A108767 in 2008, which stem from a fixed point-iteration formalism of a basic identity for a compositional inverse pair, x* h(f^{(-1)}(x)) = f^{(-1)}(x), where, as above, h(x) = x / f(x).
The sets of refined m-Narayana polynomials are used by Cachazo and Umbert to characterize the scattering amplitudes of a class of quantum fields (see, e.g., section 7.3).
These could also be called the refined 3-Dyck path polynomials. From the interpretation of A134264 as Dyck paths in A125181, or staircases whose steps never rise above the diagonal of a square grid (see illustrations in Weisstein), the monomials of the partition polynomial N_4 = 1 (4') + 4 (1') (3') + 2 (2')^2 + 6 (1')^2 (2') + 1 (1')^4 of A134264 have the following correspondences:
1 (4') --> 1 staircase of one step of height 4,
4 (1') (3') --> 4 staircases of 1 step of height 1 and 1 step of height 3,
2 (2')^2 --> 2 staircases of 2 steps of height 2,
6 (1')^2 (2') --> 6 staircases of 2 steps of height 1 and 1 step of height 2,
1 (1')^4 --> 1 staircase of 4 steps of height 1.
Consequently, the partition polynomials G_{3n} of this entry enumerate staircases of height 3n with steps of heights 3, 6, 9, ..., 3k, ... only.
Diverse combinatorial models of the refined m-Narayana, or m-Dyck, polynomials are inherited from those presented for the refined Narayana, or noncrossing partition, polynomials in A134264 and A125181 and in the references therein.
A127537 gives a combinatorial model (see title and Domb and Barret therein, Table 2, p. 355) that contains the coefficients of the monomials h_1^n and h_1^(n-2) h_2, i.e., A001764 and A003408.

			Triangle begins:
  1;
  1,  3;
  1,  9, 12;
  1, 12,  6, 66,  55;
  1, 15, 15, 105, 105, 455, 273;
  ...
Row 1: G_3  = g_3
row 2: G_6  = g_6 + 3 g_3^2
row 3: G_9  = g_9 + 9 g_3 g_6 + 12 g_3^3
row 4: G_12 = g_12 + 12 g_3 g_9 + 6 g_6^2 + 66 g_3^2 g_6 + 55 g_3^4
row 5: G_15 = g_15 + 15 g_3 g_12 + 15 g_6 g_9 + 105 g_3^2 g_9 + 105 g_3 g_6^2
              + 455 g_3^3 g_6 + 273 g_3^5.
.
In the notation of Abramowitz and Stegun p. 831 with indices of the partitions above divided by 3 and partition indeterminates h_n denoted (n):
R_1 = (1);
R_2 = (2) + 3 (1)^2;
R_3 = (3) + 9 (1) (2) + 12 (1)^3;
R_4 = (4) + 12 (1) (3) + 6 (2)^2 + 66 (1)^2 (2) + 55 (1)^4;
R_5 = (5) + 15 (1) (4) + 15 (2) (3) + 105 (1)^2 (3) + 105 (1) (2)^2 + 455 (1)^3(2)
          + 273 (1)^5.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]
F. Cachazo and B. Umbert, Connecting Scalar Amplitudes using The Positive Tropical Grassmannian, arXiv preprint arXiv:2205.02722 [hep-th], 2022.
MathOverflow, Combinatorics of iterated composition of noncrossing partition polynomials, a question posed by Tom Copeland, 2022.
Eric Weisstein's World of Mathematics, Dyck Path.

The length of row n is equal to A000041(n).
Row sums give A002293, n >= 1.
Cf. A001764, A003408, A004321, A108767, A125181, A127537, A134264, A173020, A179848, A235534, A338135.

Table[Binomial[Total[y], Length[y]-1] (Length[y]-1)! / Product[Count[y, i]!, {i, Max@@y}], {n, 8}, {y, Sort[Sort /@ IntegerPartitions[3n, n, Range[3, 3n, 3]]]}] // Flatten (* Andrey Zabolotskiy, Feb 19 2024, using Gus Wiseman's code for A134264 *)

\\ Compare with A134264
C(v)={my(n=vecsum(v), S=Set(v)); n!/((n-#v+1)!*prod(i=1, #S, my(x=S[i]); (#select(y->y==x, v))!))}
row(n)=[C(3*Vec(p)) | p<-partitions(n)]
{ for(n=1, 7, print(row(n))) } \\ Andrew Howroyd, Feb 19 2024

Coefficients of the monomials are those of the surviving monomials of the partition polynomials of A134264 after zeroing all indeterminates other than h_0, h_3, h_6, h_9, ..., h_(3n), .... The multinomial coefficients of A125181 also apply for G_n, giving the coefficient of the monomial h_1^e_1 h_2^e_2 ... h_n^n of R_n with se := e_1 + e_2 + ... + e_n as (3n)! / ((3n-se+1)! (e_1)! (e_2)! ... (e_n)!).
1*e_1 + 2*e_2 + ... + n*e_n = n for each monomial of R_n.
The partition polynomials R_n = N_n^3 of this entry can be determined from those of A338135, N_n^2, by substituting the partition polynomials of A134264, N_n, for the indeterminate h_n = (n) of N_n^2 or by doing the same for A134264 twice. E.g., N_1(h_1) = h_1, N_2(h_1,h_2) = h_2 + h_1^2, so N_2^2(h_1,h_2) = N_2(N_1,N_2) =  N_2 + N_1 = h_2 + h_1^2 + h_1^2 = h_2 + 2 h_1^2 and N_2^3(h_1,h_2) = N_2^2(N_1,N_2) = N_2 + 2 N_1^2 = h_2 + h_1^2 + 2 h_1^2 = h_2 + 3 h_1^2.
Reduces with all indeterminates h_n = (n) = t to A173020.
The coefficient of the monomial h_1^n is (3*n)! / ((3*n-n+1)! n!) = A001764(n) (see also A179848 and A235534). In general, the coefficients of these monomials of the refined (m+1)-Narayana polynomials are the Fuss-Catalan sequence associated to the row sums of the refined m-Narayana polynomials.
The coefficient of the monomial h_1^(n-2) h_2 is (3n)! / ((3n-n+2)! (n-2)!) = A003408(n-2) for n > 1.
The coefficient of the monomial h_1^(n-3) h_3 is (3n)! / ((3n-n+3)! (n-3)!) = A004321(n) for n > 2.

Rows 6-8 added by Andrey Zabolotskiy, Feb 19 2024

cp's OEIS Frontend

A002694 Binomial coefficients C(2n, n-2).

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A023822 Sum of exponents in prime-power factorization of C(3n,n-3).

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A004334 Binomial coefficient C(4n,n-4).

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A120983 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k vertices of outdegree 3 (n >= 0, k >= 0).

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A281580 a(n) = binomial(9*n, n-9).

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A354622 Irregular triangle read by rows: Refined 3-Narayana triangle. Coefficients of partition polynomials of A134264, a refined Narayana triangle enumerating noncrossing partitions, with all h_k other than h_0, h_3, h_6, ..., h_(3n), ... replaced by zero.

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