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

Author

Wolfdieter Lang, Jul 12 2001

Keywords

• nonn
• easy
• tabf

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A009766, A280759 (rows reversed).

Cf. A062746, A062747, A062748, A062749.

Cf. A005843, A000012, A000027, A000217, A005718.

Programs

Maple

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

Mathematica

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 *)

Formula

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

Extensions

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