A053792 Number of walks of length n on the square lattice that start from (0,0) and do not touch the half-line {x=y, x <= 0} once they have left their starting point.

1, 4, 10, 40, 134, 536, 1924, 7696, 28486, 113944, 429100, 1716400, 6535580, 26142320, 100308680, 401234720, 1548228166, 6192912664, 23999271964, 95997087856, 373278990004, 1493115960016, 5821831231160, 23287324924640, 91005571039516, 364022284158064

Offset

0,2

References

Mireille Bousquet-Mélou and Gilles Schaeffer, Counting walks on the slit plane (extended abstract). Mathematics and computer science (Versailles, 2000), 101-112, Trends Math., Birkhäuser, Basel, 2000.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

M. Bousquet-Mélou and Gilles Schaeffer, Walks on the slit plane, Probability Theory and Related Fields, Vol. 124, no. 3 (2002), 305-344.

Formula

G.f.: ((1+4*t)^(1/4)*(1+sqrt(1-16*t^2))^(1/2))/(sqrt(2)*(1-4*t)^(3/4)).

Contribution from Vaclav Kotesovec, Oct 24 2012: (Start)

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

Recurrence: (n-1)*n*a(n) = 8*(n-1)*a(n-1) + 4*(8*n^2-32*n+29)*a(n-2) - 128*(n-3)*a(n-3) - 256*(n-4)*(n-3)*a(n-4).

a(n) ~ 2^(2*n-1/4)/(Gamma(3/4)*n^(1/4)).

(End)

Mathematica

CoefficientList[Series[1/2*((1+4*x)/(1-4*x))^(1/4)*(1+Sqrt[(1+4*x)/(1-4*x)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

Crossrefs

Cf. A000108, A053791.

Sequence in context: A156798 A149208 A149209 * A032121 A149210 A149211

Adjacent sequences: A053789 A053790 A053791 * A053793 A053794 A053795

Keyword

nonn

Author

Mireille Bousquet-Mélou, Mar 27 2000

Status

approved