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A277359 Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1) and (1,0) on or below the diagonal and using steps (1,1), (-1,1), and (1,-1) on or above the diagonal.

%I A277359 #17 Jan 25 2017 03:06:54
%S A277359 1,2,7,32,176,1126,8227,67768,622706,6323932,70400734,852952952,
%T A277359 11176241098,157506733030,2375966883371,38200984291800,
%U A277359 652179787654530,11783182484950980,224623760504277810,4505795627243046240,94873821120923655336,2092249161797280567516
%N A277359 Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1) and (1,0) on or below the diagonal and using steps (1,1), (-1,1), and (1,-1) on or above the diagonal.
%C A277359 Both endpoints of each step have to satisfy the given restrictions.
%C A277359 a(n) is odd for n in {0, 2, 6, 14, 30, 62, ... } = { 2^n-2 | n>0 }.
%H A277359 Alois P. Heinz, <a href="/A277359/b277359.txt">Table of n, a(n) for n = 0..448</a>
%F A277359 a(n) ~ exp(1)*(exp(1)-2) * n! * n. - _Vaclav Kotesovec_, Oct 13 2016
%p A277359 a:= proc(n) option remember; `if`(n<3, [1, 2, 7][n+1],
%p A277359       ((n^3+10*n^2-10*n+1)*a(n-1)-(2*(4*n^3+2*n^2-29*n+28))
%p A277359         *a(n-2)+(4*(n-2))*(2*n-3)^2*a(n-3))/(n*(n+1)))
%p A277359     end:
%p A277359 seq(a(n), n=0..25);
%t A277359 a[n_] := a[n] = If[n<3, {1, 2, 7}[[n+1]], ((n^3+10*n^2-10*n+1)*a[n-1] - (2*(4*n^3+2*n^2-29*n+28))*a[n-2] + (4*(n-2))*(2*n-3)^2*a[n-3])/(n*(n+1)) ]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jan 25 2017, translated from Maple *)
%Y A277359 Cf. A000108, A000142, A000918, A277175, A277176, A277358, A277360, A277756.
%K A277359 nonn,walk
%O A277359 0,2
%A A277359 _Alois P. Heinz_, Oct 10 2016