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A278395
Number of positive meanders (walks starting at the origin and ending at any altitude > 0 that never touch or go below the x-axis in between) with n steps from {-3,-2,-1,1,2,3}.
6
1, 3, 12, 60, 311, 1674, 9173, 51002, 286384, 1620776, 9228724, 52810792, 303447096, 1749612736, 10117583749, 58656027314, 340806249367, 1984018271850, 11569932938192, 67574451148408, 395214184047366, 2314315680481252, 13567587349336459, 79621279809031310
OFFSET
0,2
COMMENTS
By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.
LINKS
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.
MATHEMATICA
frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -3, 3}] - 1; p = 1/x; v[[1]] = 1; For[i = 2, i <= n, i++, p = p*m // Expand; p = p - frac[p]; v[[i]] = p /. x -> 1]; v];
seq[24] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)
PROG
(PARI) seq(n)={my(v=vector(n), m=sum(i=-3, 3, x^i)-1, p=1/x); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p, x, 1)); v} \\ Andrew Howroyd, Jun 27 2018
KEYWORD
nonn,walk
AUTHOR
David Nguyen, Nov 20 2016
STATUS
approved