%I #34 Feb 08 2021 07:19:54
%S 0,0,2,6,38,130,662,2380,11174,41226,185642,695860,3055670,11576916,
%T 49995220,190876696,814610854,3128164186,13233277634,51046844836,
%U 214488337418,830382690556,3470405605900,13475470680616,56073057254198,218269673491780
%N Number of walks f length n on a square lattice ending with x > 0 and y > 0.
%H G. C. Greubel, <a href="/A186648/b186648.txt">Table of n, a(n) for n = 0..1000</a>
%F E.g.f.: ((e^(2*x)-I_0(2*x))/2)^2. - _Benjamin Phillabaum_, Mar 06 2011
%F From _Benedict W. J. Irwin_, May 25 2016: (Start)
%F If n is even, a(n) = 2^(n-2)*(2^n - 2*2F1((1-n)/2,-n/2;1;1) + n!*Gamma((n+1)/2))/(sqrt(Pi)*Gamma(1 + n/2)^3),
%F If n is odd, a(n) = 2^(n-2)*(2^n - 2*2F1((1-n)/2,-n/2;1;1)).
%F (End)
%F D-finite with recurrence n^2*(n-1)*(75*n-313)*a(n) -2*(334*n^2-1857*n+2052)*(n-1)^2*a(n-1) +8*(68*n^4-1240*n^3+6749*n^2-13464*n+9090)*a(n-2) +32*(300*n^4-2626*n^3+7387*n^2-6699*n-297)*a(n-3) -128*(218*n^2-1444*n+2429)*(-3+n)^2*a(n-4) +512*(2*n-9)*(17*n-105)*(-4+n)^2*a(n-5)=0. - _R. J. Mathar_, Feb 08 2021
%e a(3) = 6 {UUR,URU,RUU,RRU,RUR,URR}. Note: you can also go Left or Down, however that appears at the fourth sequence which is too large to put in this space.
%t Table[(CoefficientList[Series[(1/2 (E^(2 x) - (BesselI[0, 2 x])))^2, {x, 0, len}], x] Range[0, len]!)[[n + 1]], {n, 0, 25}]
%Y Cf. A187151.
%K nonn,walk
%O 0,3
%A _Benjamin Phillabaum_, Mar 06 2011