A278416 Number of meanders (walks starting at the origin and ending at any altitude >= 0 that may touch but never go below the x-axis) with n steps from {-4,-3,-2,-1,1,2,3,4}.
1, 4, 26, 174, 1231, 8899, 65492, 487646, 3664123, 27723979, 210946444, 1612394958, 12371547879, 95230159650, 735060394986, 5687343753535, 44096482961189, 342530654187820, 2665058975987628, 20765913987073659, 162019898098364055, 1265622208055843635
Offset: 0
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..200
- 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.
Programs
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Mathematica
seq[n_] := Module[{v = Table[1, n], m = Sum[ x^i, {i, -4, 4}] - 1, p = 1}, For[i = 2, i <= n, i++, p = Expand[p*m]; p = p - Select[p, Exponent[#, x] < 0&]; v[[i]] = (p /. x -> 1)]; v]; seq[25] (* Jean-François Alcover, Jul 11 2018, after Andrew Howroyd *) -
PARI
seq(n)={my(v=vector(n), m=sum(i=-4, 4, x^i)-1, p=1); v[1]=1; for(i=2, n, p*=m; p-=frac(p); v[i]=subst(p,x,1)); v} \\ Andrew Howroyd, Jun 27 2018