A190425 Number of one-sided prudent walks from (0,0) to (n,n), with floor(n/2)+n east steps, floor(n/2) west steps and n north steps.

1, 2, 12, 60, 700, 3780, 51744, 288288, 4247100, 24066900, 369881512, 2118412296, 33466634656, 193076738400, 3109838832000, 18037065225600, 294718130342460, 1716299700229620, 28355714001615000, 165657066009435000, 2761067350729888200, 16171965911417916600

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, 2010.

Formula

a(n) ~ 2^(n + 1/2) * 3^(3*n/2 - (3 - (-1)^n)/4) / (Pi*n). - Vaclav Kotesovec, Oct 21 2023

Maple

a:= n-> b(0, iquo(n, 2), n, iquo(n, 2)+n):
b:= proc(i, w, no, e) option remember; `if`(-1 in {w, no, e}, 0,
      `if`(no=0 and w=0 and e=0, 1,  b(0, w, no-1, e)+
      `if`(i<1, b(-1, w-1, no, e), 0)+`if`(i>-1, b(1, w, no, e-1), 0)))
    end:
seq(a(n), n=0..20); # Alois P. Heinz, Jun 04 2011

Mathematica

a[n_] := b[0, Quotient[n, 2], n, Quotient[n, 2] + n]; b[i_, w_, no_, e_] := b[i, w, no, e] = If[MemberQ[{w, no, e}, -1], 0, If[no == 0 && w == 0 && e == 0, 1, b[0, w, no - 1, e] + If[i < 1, b[-1, w - 1, no, e], 0] + If[i > -1, b[1, w, no, e - 1], 0]]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 24 2016, after Alois P. Heinz *)

Crossrefs

Sequence in context: A362244 A362238 A372986 * A145630 A082688 A099996

Adjacent sequences: A190422 A190423 A190424 * A190426 A190427 A190428

Keyword

nonn,walk

Author

Shanzhen Gao, May 10 2011

Extensions

More terms from Alois P. Heinz, Jun 04 2011

Status

approved