%I #20 May 14 2020 12:18:19
%S 1,2,7,38,284,2691,30890,416449,6448243,112751661,2197200541,
%T 47214026822,1109022356759,28269085769331,777140210643254,
%U 22918982645377342,721764216387297451,24173661551378798838,857993099925433301350,32168967331652245055171
%N Number of self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) such that (0,1) is never used directly before or after (1,0) or (1,1).
%H Alois P. Heinz, <a href="/A317985/b317985.txt">Table of n, a(n) for n = 0..403</a>
%F a(n) ~ c * 2^n * n! / n^(1/4), where c = 1.054816768531988358301631965137203014379828345839423725829486842843413035459... - _Vaclav Kotesovec_, May 14 2020
%p a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 38][n+1],
%p 2*n*a(n-1) -(n-2)*a(n-2) -(2*n-5)*a(n-3))
%p end:
%p seq(a(n), n=0..25);
%t a = DifferenceRoot[Function[{y, n}, {(2n+1) y[n] + (n+1) y[n+1] + (-2n-6)* y[n+2] + y[n+3] == 0, y[0] == 1, y[1] == 2, y[2] == 7, y[3] == 38}]];
%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 12 2020, after Maple *)
%t nmax = 20; CoefficientList[Simplify[Normal[Series[-1 - 1/x^(3/4) * E^(-1/(2*x) + (3*ArcTanh[(1 + 4*x)/Sqrt[17]])/(4*Sqrt[17]))* (-2 + x + 2*x^2)^(1/8) * Integrate[E^(1/(2*x)) * Simplify[Normal[Series[(-2 + 2*x + x^2)/(x^(5/4)*(-2 + x + 2*x^2)^(9/8))/ E^(3*ArcTanh[(1 + 4*x)/Sqrt[17]] / (4*Sqrt[17])), {x, 0, nmax}]], x > 0], x], {x, 0, nmax}]], x > 0], x] (* _Vaclav Kotesovec_, May 14 2020 *)
%Y Cf. A277358, A320512.
%K nonn,walk
%O 0,2
%A _Alois P. Heinz_, Oct 02 2018