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A277358 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).

%I A277358 #20 Oct 22 2018 08:58:09
%S A277358 1,2,9,58,491,5142,64159,929078,15314361,283091122,5799651689,
%T A277358 130423248378,3193954129651,84607886351462,2410542221526399,
%U A277358 73500777054712438,2388182999073694001,82374234401380995042,3006071549433968619529,115713455097715665452858
%N A277358 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).
%H A277358 Alois P. Heinz, <a href="/A277358/b277358.txt">Table of n, a(n) for n = 0..403</a>
%F A277358 E.g.f.: exp(-x/2)/(1-2*x)^(5/4).
%F A277358 a(n) = 2*n*a(n-1) + (n-1)*a(n-2) for n>1, a(0)=1, a(1)=2.
%F A277358 a(n) ~ sqrt(Pi) * 2^(n+5/2) * n^(n+3/4) / (Gamma(1/4) * exp(n+1/4)). - _Vaclav Kotesovec_, Oct 13 2016
%p A277358 a:= n-> n!*coeff(series(exp(-x/2)/(1-2*x)^(5/4), x, n+1), x, n):
%p A277358 seq(a(n), n=0..25);
%p A277358 # second Maple program:
%p A277358 a:= proc(n) option remember; `if`(n<2, n+1,
%p A277358        2*n*a(n-1) +(n-1)*a(n-2))
%p A277358     end:
%p A277358 seq(a(n), n=0..25);
%t A277358 a[n_] := a[n] = If[n < 2, n+1, 2*n*a[n-1] + (n-1)*a[n-2]];
%t A277358 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 29 2017, translated from Maple *)
%Y A277358 Cf. A277359, A277360, A277424, A284230, A317985.
%K A277358 nonn,walk
%O A277358 0,2
%A A277358 _Alois P. Heinz_, Oct 10 2016