A278394 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 {-2,-1,1,2}.

1, 2, 5, 17, 58, 209, 761, 2823, 10557, 39833, 151147, 576564, 2208163, 8486987, 32714813, 126430229, 489685674, 1900350201, 7387530575, 28763059410, 112142791763, 437774109384, 1710883748796, 6693281604018, 26210038447737, 102724200946467, 402925631267151

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, -2, 2}] - 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[27] (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

Prog

(PARI) seq(n)={my(v=vector(n), m=sum(i=-2, 2, 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, A278395, A278396, A278398.

Sequence in context: A149992 A134128 A119254 * A056679 A084528 A149993

Adjacent sequences: A278391 A278392 A278393 * A278395 A278396 A278397

Keyword

nonn,walk

Author

David Nguyen, Nov 20 2016

Status

approved