|
%I #35 Feb 13 2017 03:33:38
%S 1,2,8,26,108,382,1586,5812,24044,89846,370398,1401292,5759826,
%T 21977516,90111820,345994216,1415691244,5461770406,22308412934,
%U 86392108636,352334866238,1368640564996,5574504569620,21708901408216,88320660937298,344680279929532,1400902293406676
%N Number of walks of length n starting at origin and ending in first quadrant on a square lattice.
%H G. C. Greubel, <a href="/A187151/b187151.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..160 from Vincenzo Librandi)
%F E.g.f.: (exp(2*x)+I_0(2*x))^2/4 where I() is the Modified Bessel Function. - _Benjamin Phillabaum_, Mar 05 2011
%F Recurrence: (n-1)*n^2*(8*n^3 - 66*n^2 + 171*n - 139)*a(n) = 2*(n-1)^2*(32*n^4 - 288*n^3 + 886*n^2 - 1071*n + 396)*a(n-1) + 24*(2*n-3)*(4*n^4 - 37*n^3 + 114*n^2 - 136*n + 50)*a(n-2) - 32*(n-2)^2*(32*n^4 - 288*n^3 + 886*n^2 - 1071*n + 396)*a(n-3) + 128*(n-3)^2*(2*n-7)*(8*n^3 - 42*n^2 + 63*n - 26)*a(n-4). - _Vaclav Kotesovec_, Feb 24 2014
%F a(n) ~ 4^(n-1) * (1 + 2/sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 24 2014
%F From _Benedict W. J. Irwin_, Aug 02 2016: (Start)
%F Let b(n) = 2^(2n-2)+2^(n-1)*2F1((1-n)/2,-n/2;1;1).
%F For odd n, a(n) = b(n),
%F for even n, a(n) = b(n) + 2^(2n-2)*Gamma((n+1)/2)^2/Gamma(1+n/2)^2/Pi. (End)
%e a(2) = {UU,UR,UD,RU,RR,RL,DU,LR}.
%t CoefficientList[Series[(Exp[2x]+BesselI[0,2x])^2/4,{x,0,15}],x] * Range[0,15]!
%t Table[2^(-2 + n) (2^n + 2 Hypergeometric2F1[(1 - n)/2, -(n/2), 1, 1] + (2^n Gamma[(1 + n)/2]^2 Mod[n + 1, 2])/(Pi Gamma[1 + n/2]^2)), {n, 0, 30}] (* _Benedict W. J. Irwin_, Aug 02 2016 *)
%o (PARI) x='x+O('x^33);
%o Vec(serlaplace((exp(2*x)+besseli(0,2*x))^2/4)) /* _Joerg Arndt_, Mar 06 2011 */
%K nonn,walk,nice
%O 0,2
%A _Benjamin Phillabaum_, Mar 05 2011
|
