A189770 Number of one-sided prudent walks from (0,0) to (n,n), with 4+n east steps, 4 west steps and n north steps.

2, 30, 300, 2450, 17640, 116424, 720720, 4247100, 24066900, 132100540, 706137432, 3691172940, 18929092000, 95477508000, 474659611200, 2329787591640, 11306322135900, 54314684770500, 258550604697000, 1220631012701100, 5719528173799440, 26616851891058000

Offset

1,1

Links

Georg Fischer, Table of n, a(n) for n = 1..150 (First 42 terms from Vincenzo Librandi).

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

Formula

a(n) = (n+1)*(2+n)*(n+3)*Gamma(1+2*n) / (24*n*(Gamma(n))^2).

From Georg Fischer, Feb 25 2026: (Start)

G.f.: -2*x*(-1+3*x-6*x^2+5*x^3)/(sqrt(1-4*x)*(4*x-1)^4).

D-finite with recurrence: (n-1)*a(n)+(20*n-70)*a(n-4)+(-29*n+71)*a(n-3)+(18*n-24)*a(n-2)+(-7*n-1)*a(n-1) = 0 for n > 0. (End)

Mathematica

Table[(n+1)*(2+n)*(n+3)*Gamma[1+2*n]/(24*n*(Gamma[n])^2), {n, 30}]

Crossrefs

Sequence in context: A301350 A379557 A036351 * A245020 A277660 A089433

Adjacent sequences: A189767 A189768 A189769 * A189771 A189772 A189773

Keyword

nonn,walk

Author

Shanzhen Gao, Apr 26 2011

Status

approved