0,6

In the Frey-Sellers reference this array appears in Tab. 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 (deutsch(AT)duke.poly.edu), Jun 24 2005

D. D. Frey, J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.

D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table 2).

H. Niederhausen, Catalan traffic at the beach, The Electronic Journal of Combinatorics, 9 (2002), #R33.

W. Lang: First 10 rows.

a(0, 0)=1, a(n, -1)=0, n >= 1; a(n, r)=0 if r>2*n; a(n, r)=a(n, r-1)+a(n-1, r) else.

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(binomial(3i, i)*binomial(i-n-1, r-1-2i)/(2i+1), i=0..floor((r-1)/2)) (0<=r<=2n) (see the Niederhausen reference, eq. (17)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 24 2005

{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.

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 (Deutsch)

Cf. A009766.

Sequence in context: A016738 A061911 A082239 this_sequence A140733 A143605 A098418

Adjacent sequences: A062742 A062743 A062744 this_sequence A062746 A062747 A062748

nonn,easy,tabf

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 12 2001

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