A278396 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 {-4,-3,-2,-1,1,2,3,4}.

1, 4, 22, 146, 1013, 7269, 53156, 394154, 2951950, 22279439, 169175927, 1290970376, 9891573310, 76050920691, 586426828071, 4533349152056, 35122039919110, 272634162463779, 2119948044144136, 16509519223752380, 128747868290672353, 1005273235488567875

Offset

0,2

Comments

By convention, the empty walk (corresponding to n=0) is considered to be a positive meander.

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.

Mathematica

frac[ex_] := Select[ex, Exponent[#, x] < 0&];
seq[n_] := Module[{v, m, p}, v = Table[0, n]; m = Sum[x^i, {i, -4, 4}] - 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[22] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

Prog

(PARI) seq(n)={my(v=vector(n), m=sum(i=-4, 4, 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

Crossrefs

Cf. A276852, A278391, A278392, A278393, A278394, A278395, A278398.

Sequence in context: A222012 A057834 A196795 * A369617 A121394 A005039

Adjacent sequences: A278393 A278394 A278395 * A278397 A278398 A278399

Keyword

nonn,walk

Author

David Nguyen, Nov 20 2016

Status

approved