A280759 -id:A280759 - cp's OEIS frontend

A062745 Generalized Catalan array FS(3; n,r).

1, 1, 1, 1, 1, 2, 3, 3, 3, 1, 3, 6, 9, 12, 12, 12, 1, 4, 10, 19, 31, 43, 55, 55, 55, 1, 5, 15, 34, 65, 108, 163, 218, 273, 273, 273, 1, 6, 21, 55, 120, 228, 391, 609, 882, 1155, 1428, 1428, 1428, 1, 7, 28, 83, 203, 431, 822, 1431, 2313, 3468, 4896, 6324, 7752, 7752

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this array appears in Table 2, p. 143 and is called {n over r}_{m-1}, with m=3.
The step width sequence of this staircase array is [1,2,2,2,....], i.e., the degree of the row polynomials is [0,2,4,6,...] = A005843.
The columns r=0..5 give A000012 (powers of 1), A000027 (natural), A000217 (triangular), A062748, A005718, A062749.
Number of lattice paths from (0,0) to (r,n) using steps h=(1,0), v=(0,1) and staying on or above the line y = x/2. Example: a(3,2)=6 because from (0,0) to (2,3) we have the following valid paths: vvvhh, vvhvh, vvhhv, vhvvh, vhvhvh and vhvvh (see the Niederhausen reference). - Emeric Deutsch, Jun 24 2005

			Array begins:
  {1};
  {1,1,1};
  {1,2,3,3,3};
  {1,3,6,9,12,12,12};
  ...;
N(3; 1,x) = 3-3*x+x^2.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
Wolfdieter Lang, First 10 rows.
Toufik Mansour and I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table 2).
Heinrich Niederhausen, Catalan Traffic at the Beach, Electronic Journal of Combinatorics, Volume 9 (2002), #R33.

Cf. A009766, A280759 (rows reversed).
Cf. A062746, A062747, A062748, A062749.
Cf. A005843, A000012, A000027, A000217, A005718.

a:=proc(n,r) if r<=2*n then binomial(n+r,r)-(-1)^(r-1)*sum(binomial(3*i,i)*binomial(i-n-1,r-1-2*i)/(2*i+1),i=0..floor((r-1)/2)) else 0 fi end: for n from 0 to 8 do seq(a(n,r),r=0..2*n) od; # yields sequence in triangular form # Emeric Deutsch, Jun 24 2005

a[0, 0] = 1; a[, -1] = 0; a[n, r_] /; r > 2*n = 0; a[n_, r_] := a[n, r] = a[n, r-1] + a[n-1, r]; Table[a[n, r], {n, 0, 7}, {r, 0, 2*n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

a(0,0)=1, a(n,-1)=0, n >= 1; a(n,r) = a(n, r-1) + a(n-1, r) if r <= 2n, 0 otherwise.
G.f. for column r = 2*k+j, k >= 0, j=1, 2: (x^(k+1))*N(3; k, x)/ (1-x)^(2*k+1+j), with row polynomials N(3; k, x) of array A062746; for column r=0: 1/(1-x).
a(n,r) = binomial(n+r, r) - (-1)^(r-1)*Sum_{i=0..floor((r-1)/2)} binomial(3i, i)*binomial(i-n-1, r-1-2i)/(2i+1), 0 <= r <= 2n (see the Niederhausen reference, eq. (17)). - Emeric Deutsch, Jun 24 2005

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

A334680 a(n) is the total number of down-steps after the final up-step in all 2-Dyck paths of length 3n (n up-steps and 2n down-steps).

Original entry on oeis.org

0, 2, 9, 43, 218, 1155, 6324, 35511, 203412, 1184040, 6983925, 41652468, 250763464, 1521935948, 9301989144, 57203999295, 353701790376, 2197600497330, 13713291247635, 85907187607395, 540072341320050, 3406202392821375, 21545888897092560, 136655834260685220, 868897745157965328

Offset: 0

Andrei Asinowski, May 08 2020

A 2-Dyck path is a lattice path with steps U = (1, 2), d = (1, -1) that starts at (0,0), stays (weakly) above the x-axis, and ends at the x-axis.

			For n = 2, the a(2) = 9 is the total number of down-steps after the last up-step in UddUdd, UdUddd, UUdddd.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

First order differences of A001764.
The 4th column of A280759.
Cf. A062745.

alias(PS=ListTools:-PartialSums): A334680List := proc(m) local A, P, n;
A := [0,2]; P := [1,2]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-1]] od; A end: A334680List(25); # Peter Luschny, Mar 26 2022

a[n_] := Binomial[3*n + 4, n + 1]/(3*n + 4) - Binomial[3*n + 1, n]/(3*n + 1); Array[a, 25, 0] (* Amiram Eldar, May 13 2020 *)

[(17 + 23*n)*binomial(3*n, n-1)/(2*n+2)/(2*n+3) for n in srange(30)] # Benjamin Hackl, May 13 2020

a(n) = binomial(3*(n+1) + 1, n+1)/(3*(n+1) + 1) - binomial(3*n + 1, n)/(3*n + 1).
a(n) = (17 + 23*n)*binomial(3*n, n - 1)/((2*n + 2)*(2*n + 3)).
a(n) = A062745(n+1, 2*n-1).

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A062745 Generalized Catalan array FS(3; n,r).

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A334680 a(n) is the total number of down-steps after the final up-step in all 2-Dyck paths of length 3n (n up-steps and 2n down-steps).

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cp's OEIS Frontend

A062745 Generalized Catalan array FS(3; n,r).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A334680 a(n) is the total number of down-steps after the final up-step in all 2-Dyck paths of length 3*n (n up-steps and 2*n down-steps).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A334680 a(n) is the total number of down-steps after the final up-step in all 2-Dyck paths of length 3n (n up-steps and 2n down-steps).