A004318 -id:A004318 - cp's OEIS frontend

A002054 Binomial coefficient C(2n+1, n-1).

1, 5, 21, 84, 330, 1287, 5005, 19448, 75582, 293930, 1144066, 4457400, 17383860, 67863915, 265182525, 1037158320, 4059928950, 15905368710, 62359143990, 244662670200, 960566918220, 3773655750150, 14833897694226, 58343356817424, 229591913401900

Offset: 1

N. J. A. Sloane

a(n) = number of permutations in S_{n+2} containing exactly one 312 pattern. E.g., S_3 has a_1 = 1 permutations containing exactly one 312 pattern, and S_4 has a_2 = 5 permutations containing exactly one 312 pattern, namely 1423, 2413, 3124, 3142, and 4231. This comment is also true if 312 is replaced by any of 132, 213, or 231 (but not 123 or 321, for which see A003517). [Comment revised by N. J. A. Sloane, Nov 26 2022]
Number of valleys in all Dyck paths of semilength n+1. Example: a(2)=5 because UD*UD*UD, UD*UUDD, UUDD*UD, UUD*UDD, UUUDDD, where U=(1,1), D=(1,-1) and the valleys are shown by *. - Emeric Deutsch, Dec 05 2003
Number of UU's (double rises) in all Dyck paths of semilength n+1. Example: a(2)=5 because UDUDUD, UDU*UDD, U*UDDUD, U*UDUDD, U*U*UDDD, the double rises being shown by *. - Emeric Deutsch, Dec 05 2003
Number of peaks at level higher than one (high peaks) in all Dyck paths of semilength n+1. Example: a(2)=5 because UDUDUD, UDUU*DD, UU*DDUD, UU*DU*DD, UUU*DDD, the high peaks being shown by *. - Emeric Deutsch, Dec 05 2003
Number of diagonal dissections of a convex (n+3)-gon into n regions. Number of standard tableaux of shape (n,n,1) (see Stanley reference). - Emeric Deutsch, May 20 2004
Number of dissections of a convex (n+3)-gon by noncrossing diagonals into several regions, exactly n-1 of which are triangular. Example: a(2)=5 because the convex pentagon ABCDE is dissected by any of the diagonals AC, BD, CE, DA, EB into regions containing exactly 1 triangle. - Emeric Deutsch, May 31 2004
Number of jumps in all full binary trees with n+1 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. - Emeric Deutsch, Jan 18 2007
a(n) is the total number of nonempty Dyck subpaths in all Dyck paths (A000108) of semilength n. For example, the Dyck path UUDUUDDD has Dyck subpaths stretching over positions 1-8 (the entire path), 2-3, 2-7, 4-7, 5-6 and so contributes 5 to a(4). - David Callan, Jul 25 2008
a(n+1) is the total number of ascents in the set of all n-permutations avoiding the pattern 132. For example, a(2) = 5 because there are 5 ascents in the set 123, 213, 231, 312, 321. - Cheyne Homberger, Oct 25 2013
Number of increasing tableaux of shape (n+1,n+1) with largest entry 2n+1. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. Example: a(2) = 5 counts the five tableaux (124)(235), (123)(245), (124)(345), (134)(245), (123)(245). - Oliver Pechenik, May 02 2014
a(n) is the number of noncrossing partitions of 2n+1 into n-1 blocks of size 2 and 1 block of size 3. - Oliver Pechenik, May 02 2014
Number of paths in the half-plane x>=0, from (0,0) to (2n+1,3), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=2, we have the 5 paths: UUUUD, UUUDU, UUDUU, UDUUU, DUUUU. - José Luis Ramírez Ramírez, Apr 19 2015
From Gus Wiseman, Aug 20 2021: (Start)
Also the number of binary numbers with 2n+2 digits and with two more 0's than 1's. For example, the a(2) = 5 binary numbers are: 100001, 100010, 100100, 101000, 110000, with decimal values 33, 34, 36, 40, 48. Allowing first digit 0 gives A001791, ranked by A345910/A345912.
Also the number of integer compositions of 2n+2 with alternating sum -2, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(3) = 21 compositions are:
  (35)  (152)  (1124)  (11141)  (111113)
        (251)  (1223)  (12131)  (111212)
               (1322)  (13121)  (111311)
               (1421)  (14111)  (121112)
               (2114)           (121211)
               (2213)           (131111)
               (2312)
               (2411)
The following pertain to these compositions:
- The unordered version is A344741.
- Ranked by A345924 (reverse: A345923).
- A345197 counts compositions by length and alternating sum.
- A345925 ranks compositions with alternating sum 2 (reverse: A345922).
(End)

			G.f. = x + 5*x^2 + 21*x^3 + 84*x^4 + 330*x^5 + 1287*x^6 + 5005*x^7 + ...

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.
George Grätzer, General Lattice Theory. Birkhauser, Basel, 1998, 2nd edition, p. 474, line -3.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
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).

G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..100 computed by T. D. Noe)
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].
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
F. R. Bernhart and N. J. A. Sloane, Emails, April-May 1994.
Jean-Luc Baril and Sergey Kirgizov, The pure descent statistic on permutations, Discrete Mathematics, Vol. 340, No. 10 (2017), pp. 2550-2558; preprint, 2017.
Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.
David Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
Arthur Cayley, On the partitions of a polygon, Proc. London Math. Soc., Vol. 22 (1891), pp. 237-262 = Collected Mathematical Papers, Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020.
Colin Defant, Proofs of Conjectures about Pattern-Avoiding Linear Extensions, arXiv:1905.02309 [math.CO], 2019.
Emeric Deutsch, Dyck path enumeration, Discrete Math., Vol. 204, No. 1-3 (1999), pp. 167-202.
Gennady Eremin, Dyck Numbers, IV. Nested patterns in OEIS A036991, arXiv:2306.10318 [math.CO], 2023. See (5) p. 7.
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, J. Int. Seq., Vol. 17 (2014), Article 14.1.5; arXiv preprint, arXiv:1203.6792 [math.CO], 2012.
Xiaoyu He, Emily Huang, Ihyun Nam and Rishubh Thaper, Shuffle Squares and Reverse Shuffle Squares, arXiv:2109.12455 [math.CO], 2021.
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
Milan Janjic, Two Enumerative Functions.
Werner Krandick, Trees and jumps and real roots, J. Computational and Applied Math., Vol. 162, No. 1 (2004), pp. 51-55.
Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5.
Toufik Mansour and Alek Vainshtein, Counting occurrences of 123 in a permutation, arXiv:math/0105073 [math.CO], 2001.
Henri Mühle, Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices, arXiv:1509.06942 [math.CO], 2015.
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, Vol. 15 (2012), Article 12.3.3.
John Noonan and Doron Zeilberger, The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns, arXiv:math/9808080 [math.CO], 1998.
Oliver Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, arXiv:1209.1355 [math.CO], 2012-2014.
Oliver Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory A, Vol. 125 (2014), pp. 357-378.
Karol Penson, Hausdorff moment problems for combinatorial numbers: heuristics via Meijer G-functions, Nov. 2022.
Ronald C. Read, On general dissections of a polygon, Aequationes Mathematicae, Vol. 18, No. 1-2 (1978), pp. 370-388; Preprint, 1974.
Mark Shattuck, Enumeration of non-crossing partitions according to subwords with repeated letters, arXiv:2303.06300 [math.CO], 2023.
Richard P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, Vol. 76 , No. 1 (1996), pp. 175-177.
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).
A. Vogt, Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation, arXiv:1108.2993 [hep-ph], 2011.
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.

Diagonal 4 of triangle A100257. Also a diagonal of A033282.
Equals (1/2) A024483(n+2). Bisection of A037951 and A037955.
Cf. A001263.
Column k=1 of A263771.
Counts terms of A031445 with 2n+2 digits in binary.
Cf. A000097, A000346, A000984, A001622, A001700, A007318, A008549, A031444, A058622, A097805, A116406, A138364, A163493, A202736.
Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002694 (m = 4), A003516 (m = 5), A002696 (m = 6), A030053 - A030056, A004310 - A004318.

List([1..25],n->Binomial(2*n+1,n-1)); # Muniru A Asiru, Aug 09 2018

[Binomial(2*n+1, n-1): n in [1..30]]; // Vincenzo Librandi, Apr 20 2015

with(combstruct): seq((count(Composition(2*n+2), size=n)), n=1..24); # Zerinvary Lajos, May 03 2007

CoefficientList[Series[8/(((Sqrt[1-4x] +1)^3)*Sqrt[1-4x]), {x,0,22}], x] (* Robert G. Wilson v, Aug 08 2011 *)
a[ n_]:= Binomial[2 n + 1, n - 1]; (* Michael Somos, Apr 25 2014 *)

PARI
```
{a(n) = binomial( 2*n+1, n-1)};
```

from _future_ import division
A002054_list, b = [], 1
for n in range(1,10**3):
    A002054_list.append(b)
    b = b*(2*n+2)*(2*n+3)//(n*(n+3)) # Chai Wah Wu, Jan 26 2016

[binomial(2*n+1, n-1) for n in (1..25)] # G. C. Greubel, Mar 22 2019

a(n) = Sum_{j=0..n-1} binomial(2*j, j) * binomial(2*n - 2*j, n-j-1)/(j+1). - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
G.f.: z*C^4/(2-C), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. - Emeric Deutsch, Jul 05 2003
From Wolfdieter Lang, Jan 09 2004: (Start)
a(n) = binomial(2*n+1, n-1) = n*C(n+1)/2, C(n)=A000108(n) (Catalan).
G.f.: (1 - 2*x - (1-3*x)*c(x))/(x*(1-4*x)) with g.f. c(x) of A000108. (End)
G.f.: z*C(z)^3/(1-2*z*C(z)), where C(z) is the g.f. of Catalan numbers. - José Luis Ramírez Ramírez, Apr 19 2015
G.f.: 2F1(5/2, 2; 4; 4*x). - R. J. Mathar, Aug 09 2015
D-finite with recurrence: a(n+1) = a(n)*(2*n+3)*(2*n+2)/(n*(n+3)). - Chai Wah Wu, Jan 26 2016
From Ilya Gutkovskiy, Aug 30 2016: (Start)
E.g.f.: (BesselI(0,2*x) + (1 - 1/x)*BesselI(1,2*x))*exp(2*x).
a(n) ~ 2^(2*n+1)/sqrt(Pi*n). (End)
a(n) = (1/(n+1))*Sum_{i=0..n-1} (n+1-i)*binomial(2n+2,i), n >= 1. - Taras Goy, Aug 09 2018
G.f.: (x - 1 + (1 - 3*x)/sqrt(1 - 4*x))/(2*x^2). - Michael Somos, Jul 28 2021
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=1} 1/a(n) = 5/3 - 2*Pi/(9*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 52*log(phi)/(5*sqrt(5)) - 7/5, where phi is the golden ratio (A001622). (End)
a(n) = A001405(2*n+1) - A000108(n+1), n >= 1 (from Eremin link, page 7). - Gennady Eremin, Sep 05 2023
G.f.: x/(1 - 4*x)^2 * c(-x/(1 - 4*x))^3, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Feb 03 2024
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) * sqrt(x)*(x - 3)/sqrt(4 - x) (see Penson).
G.f. x*/sqrt(1 - 4*x) * c(x)^3. (End)

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

Original entry on oeis.org

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)

A003516 Binomial coefficients C(2n+1, n-2).

Original entry on oeis.org

1, 7, 36, 165, 715, 3003, 12376, 50388, 203490, 817190, 3268760, 13037895, 51895935, 206253075, 818809200, 3247943160, 12875774670, 51021117810, 202112640600, 800472431850, 3169870830126, 12551759587422

Offset: 2

N. J. A. Sloane

a(n) is the number of royal paths (A006318) from (0,0) to (n,n) with exactly one diagonal step off the line y=x. - David Callan, Mar 25 2004

a(n) is the total number of DDUU's in all Dyck (n+2)-paths. - David Scambler, May 03 2013

			For n=4, C(2*4+1,4-2) = C(9,2) = 9*8/2 = 36, so a(4) = 36. - _Michael B. Porter_, Sep 10 2016

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 2..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].
Heidi Goodson, An Identity for Vertically Aligned Entries in Pascal's Triangle, arXiv:1901.08653 [math.CO], 2019.
Milan Janjic, Two Enumerative Functions.
Toufik Mansour and Mark Shattuck, Counting occurrences of subword patterns in non-crossing partitions, Art Disc. Appl. Math. (2022).
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, Vol. 15 (2012), Article 12.3.3. - From _N. J. A. Sloane_, Sep 16 2012
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).

Diagonal 6 of triangle A100257.
Third unsigned column (s=2) of A113187. - Wolfdieter Lang, Oct 18 2012
Cf. triangle A114492 - Dyck paths with k DDUU's.
Cf. A001622, A115139.
Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A002696 (m = 6), A030053 - A030056, A004310 - A004318.

List([2..25], n-> Binomial(2*n+1, n-2)); # G. C. Greubel, Mar 21 2019

[Binomial(2*n+1,n-2): n in [2..25]]; // Vincenzo Librandi, Apr 13 2011

CoefficientList[ Series[ 32/(((Sqrt[1 - 4 x] + 1)^5)*Sqrt[1 - 4 x]), {x, 0, 25}], x] (* Robert G. Wilson v, Aug 08 2011 *)
Table[Binomial[2*n +1,n-2], {n,2,25}] (* G. C. Greubel, Jan 23 2017 *)

{a(n) = binomial(2*n+1, n-2)}; \\ G. C. Greubel, Mar 21 2019

[binomial(2*n+1, n-2) for n in (2..25)] # G. C. Greubel, Mar 21 2019

G.f.: 32*x^2/(sqrt(1-4*x)*(sqrt(1-4*x)+1)^5). - Marco A. Cisneros Guevara, Jul 18 2011
a(n) = Sum_{k=0..n-2} binomial(n+k+2,k). - Arkadiusz Wesolowski, Apr 02 2012
D-finite with recurrence (n+3)*(n-2)*a(n) = 2*n*(2*n+1)*a(n-1). - R. J. Mathar, Oct 13 2012
G.f.: x^2*c(x)^5/sqrt(1-4*x) = ((-1 + 2*x) + (1 - 3*x + x^2) * c(x))/(x^2*sqrt(1-4*x)), with c(x) the o.g.f. of the Catalan numbers A000108. See the W. Lang link under A115139 for powers of c. - Wolfdieter Lang, Sep 10 2016
a(n) ~ 2^(2*n+1)/sqrt(Pi*n). - Ilya Gutkovskiy, Sep 10 2016
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=2} 1/a(n) = 4 - 14*Pi/(9*sqrt(3)).
Sum_{n>=2} (-1)^n/a(n) = 228*log(phi)/(5*sqrt(5)) - 134/15, where phi is the golden ratio (A001622). (End)
G.f.: 2F1([7/2,3],[6],4*x). - Karol A. Penson, Apr 24 2024
a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * sqrt(x)*(x^2 - 5*x + 5)/sqrt(4 - x). - Peter Bala, Oct 13 2024

A002696 Binomial coefficients C(2n,n-3).

Original entry on oeis.org

1, 8, 45, 220, 1001, 4368, 18564, 77520, 319770, 1307504, 5311735, 21474180, 86493225, 347373600, 1391975640, 5567902560, 22239974430, 88732378800, 353697121050, 1408831480056, 5608233007146, 22314239266528, 88749815264600, 352870329957600, 1402659561581460

Offset: 3

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=3. - Herbert Kociemba, May 23 2004

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).

Robert Israel, Table of n, a(n) for n = 3..1497
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].
A. Claesson and T. Mansour, Counting patterns of type (1,2) or (2,1), arXiv:math/0110036 [math.CO], 2001.
Milan Janjic, Two Enumerative Functions
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
Toufik Mansour and Mark Shattuck, Counting occurrences of subword patterns in non-crossing partitions, Art Disc. Appl. Math. (2022).
R. Parviainen, Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Hermann Stamm-Wilbrandt, Compute C(2n, n-k) based on C(n,...) animation
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).

Diagonal 7 of triangle A100257.
Column k=1 of A263776.
Cf. A001622.
Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A003516 (m = 5), A030053 - A030056, A004310 - A004318.

List([3..30], n-> Binomial(2*n, n-3)) # G. C. Greubel, Mar 21 2019

[ Binomial(2*n,n-3): n in [3..30] ]; // Vincenzo Librandi, Apr 13 2011

A002696:=n->binomial(2*n,n-3): seq(A002696(n), n=3..30); # Wesley Ivan Hurt, Aug 19 2015

CoefficientList[Series[64/(((Sqrt[1-4x] +1)^6)*Sqrt[1-4x]), {x,0,30}], x] (* Robert G. Wilson v, Aug 08 2011 *)

a(n)=binomial(n+n,n-3) \\ Charles R Greathouse IV, Aug 08 2011

[binomial(2*n, n-3) for n in (3..30)] # G. C. Greubel, Mar 21 2019

G.f.: (1-sqrt(1-4*z))^6/(64*z^3*sqrt(1-4*z)). - Emeric Deutsch, Jan 28 2004
a(n) = Sum_{k=0..n} C(n, k)*C(n, k+3). - Hermann Stamm-Wilbrandt, Aug 17 2015
From Robert Israel, Aug 19 2015: (Start)
(n-2)*(n+4)*a(n+1) = (2*n+2)*(2*n+1)*a(n).
E.g.f.: I_3(2*x) * exp(2*x) where I_3 is a modified Bessel function. (End)
From Amiram Eldar, Aug 27 2022: (Start)
Sum_{n>=3} 1/a(n) = 3/4 + 2*Pi/(9*sqrt(3)).
Sum_{n>=3} (-1)^(n+1)/a(n) = 444*log(phi)/(5*sqrt(5)) - 1093/60, where phi is the golden ratio (A001622). (End)
G.f.: 2F1([7/2,4],[7],4*x). - Karol A. Penson, Apr 24 2024
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^3 - 6*x^2 + 9*x - 2)/sqrt(x*(4 - x)).
G.f: x^3 * B(x) * C(x)^6, 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)

More terms from Emeric Deutsch, Feb 18 2004

A030053 a(n) = binomial(2n+1,n-3).

Original entry on oeis.org

1, 9, 55, 286, 1365, 6188, 27132, 116280, 490314, 2042975, 8436285, 34597290, 141120525, 573166440, 2319959400, 9364199760, 37711260990, 151584480450, 608359048206, 2438362177020, 9762479679106, 39049918716424, 156077261327400, 623404249591760

Offset: 3

N. J. A. Sloane

Number of UUUUUU's in all Dyck (n+3)-paths. - David Scambler, May 03 2013

			G.f. = x^3 + 9*x^4 + 55*x^5 + 286*x^6 + 1365*x^7 + 6188*x68 + ...

T. D. Noe, Table of n, a(n) for n = 3..200
Milan Janjic, Two Enumerative Functions.
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, Vol. 15 (2012), Article 12.3.3. - From _N. J. A. Sloane_, Sep 16 2012.

Diagonal 8 of triangle A100257.
Cf. A001622, A113187 (unsigned fourth column).
Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A003516 (m = 5), A002696 (m = 6), A030054 - A030056, A004310 - A004318.

[Binomial(2*n+1,n-3): n in [3..30]]; // Vincenzo Librandi, Aug 11 2015

Table[Binomial[2*n + 1, n - 3], {n, 3, 20}] (* T. D. Noe, Apr 03 2014 *)
Rest[Rest[Rest[CoefficientList[Series[128 x^3 / ((1 - Sqrt[1 - 4 x])^7 Sqrt[1 - 4 x]) + (-1 / x^4 + 5 / x^3 - 6 / x^2 + 1 / x), {x, 0, 40}], x]]]] (* Vincenzo Librandi, Aug 11 2015 *)

a(n) = binomial(2*n+1,n-3); \\ Joerg Arndt, May 08 2013

G.f.: x^3*128/((1-sqrt(1-4*x))^7*sqrt(1-4*x))+(-1/x^4+5/x^3-6/x^2+1/x). - Vladimir Kruchinin, Aug 11 2015
D-finite with recurrence: -(n+4)*(n-3)*a(n) +2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 28 2020
G.f.: x^3* 2F1(4,9/2;8;4x). - R. J. Mathar, Jan 28 2020
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=3} 1/a(n) = 22*Pi/(9*sqrt(3)) - 33/10.
Sum_{n>=3} (-1)^(n+1)/a(n) = 852*log(phi)/(5*sqrt(5)) - 1073/30, 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)) * sqrt(x)*(x^3 - 7*x^2 + 14*x - 7)/sqrt((4 - x)).
G.f. x^3 * B(x) * C(x)^7, 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)

cp's OEIS Frontend

A002054 Binomial coefficient C(2n+1, n-1).

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

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A003516 Binomial coefficients C(2n+1, n-2).

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A002696 Binomial coefficients C(2n,n-3).

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A030053 a(n) = binomial(2n+1,n-3).

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