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A277424 Total number of nodes summed over all 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 A277424 #23 Jun 19 2017 08:20:11
%S A277424 1,5,39,379,4457,61503,974107,17412317,346662981,7605810685,
%T A277424 182298744203,4738700778123,132767583248917,3988244997744743,
%U A277424 127859570155253607,4357113615504651565,157266354405499307369,5993377455733610208885,240479249123008267155343
%N A277424 Total number of nodes summed over all 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 A277424 Alois P. Heinz, <a href="/A277424/b277424.txt">Table of n, a(n) for n = 0..400</a>
%F A277424 From _Vaclav Kotesovec_, Oct 14 2016: (Start)
%F A277424 Recurrence: (n^3 + n^2 - 6*n + 1)*a(n) = (4*n^4 + 6*n^3 - 28*n^2 + 3*n + 4)*a(n-1) - (4*n^5 + 2*n^4 - 42*n^3 + 50*n^2 - 13)*a(n-2) - (n-2)*(4*n^4 + 8*n^3 - 34*n^2 - n + 16)*a(n-3) - (n-3)*(n-2)*(n^3 + 4*n^2 - n - 3)*a(n-4).
%F A277424 a(n) ~ sqrt(Pi) * 2^(n+3/2) * n^(n+11/4) / (3 * Gamma(1/4) * exp(n+1/4)).
%F A277424 (End)
%p A277424 b:= proc(x, y, t) option remember; `if`(x<0 or y<0, 0,
%p A277424       `if`(x=0 and y=0, [1$2], (p-> p+ [0, p[1]])(
%p A277424        b(x-1, y, 0)+ b(x, y-1, 0)+ b(x-1, y-1, 0)+
%p A277424       `if`(t<>2, b(x+1, y-1, 1), 0)+
%p A277424       `if`(t<>1, b(x-1, y+1, 2), 0))))
%p A277424     end:
%p A277424 a:= n-> b(n, 0$2)[2]:
%p A277424 seq(a(n), n=0..25);
%t A277424 b[x_, y_, t_] := b[x, y, t] = If[x < 0 || y < 0, 0, If[x == 0 && y == 0, {1, 1}, Function[p, p + {0, p[[1]]}][b[x - 1, y, 0] + b[x, y - 1, 0] + b[x - 1, y - 1, 0] + If[t != 2, b[x + 1, y - 1, 1], 0] + If[t != 1, b[x - 1, y + 1, 2], 0]]]];
%t A277424 a[n_] := b[n, 0, 0][[2]];
%t A277424 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jun 19 2017, translated from Maple *)
%Y A277424 Cf. A277358, A284231, A285673.
%K A277424 nonn
%O A277424 0,2
%A A277424 _Alois P. Heinz_, Oct 14 2016