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%I #35 Jun 23 2023 07:48:36
%S 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,
%T 34,65,108,163,218,273,273,273,1,6,21,55,120,228,391,609,882,1155,
%U 1428,1428,1428,1,7,28,83,203,431,822,1431,2313,3468,4896,6324,7752,7752
%N Generalized Catalan array FS(3; n,r).
%C In the Frey-Sellers reference this array appears in Table 2, p. 143 and is called {n over r}_{m-1}, with m=3.
%C 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.
%C The columns r=0..5 give A000012 (powers of 1), A000027 (natural), A000217 (triangular), A062748, A005718, A062749.
%C 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
%H Vincenzo Librandi, <a href="/A062745/b062745.txt">Table of n, a(n) for n = 0..1000</a>
%H D. D. Frey and J. A. Sellers, <a href="http://www.fq.math.ca/Scanned/39-2/frey.pdf">Generalizing Bailey's generalization of the Catalan numbers</a>, The Fibonacci Quarterly, 39 (2001) 142-148.
%H Wolfdieter Lang, <a href="/A062745/a062745.txt">First 10 rows.</a>
%H Toufik Mansour and I. L. Ramirez, <a href="https://ajc.maths.uq.edu.au/pdf/81/ajc_v81_p447.pdf">Enumerations of polyominoes determined by Fuss-Catalan words</a>, Australas. J. Combin. 81 (3) (2021) 447-457.
%H D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.1006/jcta.2002.3273">The tennis ball problem</a>, J. Combin. Theory, A 99 (2002), 307-344 (Table 2).
%H Heinrich Niederhausen, <a href="https://doi.org/10.37236/1649">Catalan Traffic at the Beach</a>, Electronic Journal of Combinatorics, Volume 9 (2002), #R33.
%F 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.
%F 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).
%F 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
%e Array begins:
%e {1};
%e {1,1,1};
%e {1,2,3,3,3};
%e {1,3,6,9,12,12,12};
%e ...;
%e N(3; 1,x) = 3-3*x+x^2.
%p 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
%t 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 *)
%Y Cf. A009766, A280759 (rows reversed).
%Y Cf. A062746, A062747, A062748, A062749.
%Y Cf. A005843, A000012, A000027, A000217, A005718.
%K nonn,easy,tabf
%O 0,6
%A _Wolfdieter Lang_, Jul 12 2001
%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
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