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 A292437 a(n) is the number of lattice walks from (0,0) to (3*n,3*n) that use steps in directions {(3,0), (2,1), (1,2), (0,3)} and stay weakly below the line y=x. 1
 1, 2, 13, 120, 1288, 15046, 185658, 2380720, 31411376, 423660504, 5814905977, 80956085304, 1140478875656, 16227516683124, 232870988052180, 3366482778363616, 48981220255732960, 716707681487535144, 10539913681632290532, 155697664218428455520, 2309297999296926348448 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..834 Jackson Evoniuk, Steven Klee, Van Magnan, Enumerating Minimal Length Lattice Paths, 2017, also Enumerating Minimal Length Lattice Paths, J. Int. Seq., Vol. 21 (2018), Article 18.3.6. EXAMPLE For n=2, the a(2)=13 paths terminating at (6,6) are (3, 0), (3, 0), (0, 3), (0, 3) (3, 0), (2, 1), (1, 2), (0, 3) (3, 0), (2, 1), (0, 3), (1, 2) (3, 0), (1, 2), (2, 1), (0, 3) (3, 0), (1, 2), (1, 2), (1, 2) (3, 0), (0, 3), (3, 0), (0, 3) (3, 0), (0, 3), (2, 1), (1, 2) (2, 1), (3, 0), (1, 2), (0, 3) (2, 1), (3, 0), (0, 3), (1, 2) (2, 1), (2, 1), (2, 1), (0, 3) (2, 1), (2, 1), (1, 2), (1, 2) (2, 1), (1, 2), (3, 0), (0, 3) (2, 1), (1, 2), (2, 1), (1, 2) MAPLE b:= proc(l) option remember; `if`(l=[0\$2], 1, add(       (f-> `if`(min(f)<0 or f[1] b([3*n\$2]): seq(a(n), n=0..25);  # Alois P. Heinz, Dec 09 2017 PROG (Sage) S = [[3, 0], [2, 1], [1, 2], [0, 3]] q = 10 numPathsMat = matrix(q+1, q+1, 0) for m in [0..q]: ....for n in [0..m]: ........count = 0 ........for s in S: ............if n-s[1]>=0 and m-s[0]>=n-s[1]: ................count += numPathsMat[m-s[0], n-s[1]] ........numPathsMat[m, n] = count ........numPathsMat[0, 0] = 1 print numPathsMat.diagonal() CROSSREFS Cf. A000108, A007318. Sequence in context: A209217 A000180 A215715 * A317196 A192460 A004122 Adjacent sequences:  A292434 A292435 A292436 * A292438 A292439 A292440 KEYWORD nonn,walk AUTHOR Steven Klee, Dec 08 2017 STATUS approved

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Last modified October 23 22:23 EDT 2019. Contains 328373 sequences. (Running on oeis4.)