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A284778 Number of self-avoiding planar walks of length n+1 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) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal.

%I A284778 #34 Mar 21 2023 04:10:41
%S A284778 0,1,1,4,8,22,54,142,370,983,2627,7086,19238,52561,144377,398518,
%T A284778 1104794,3074809,8588093,24064642,67630898,190584766,538412426,
%U A284778 1524554956,4326119748,12300296227,35037658099,99977847308,285741659312,817901027070,2344475178110
%N A284778 Number of self-avoiding planar walks of length n+1 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) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal.
%C A284778 From _Gus Wiseman_, Nov 15 2022: (Start)
%C A284778 Conjecture: Also the number of topologically series-reduced ordered rooted trees with n + 3 vertices and more than one branch of the root. This would imply a(n) = A187306(n+1) - A005043(n+1). For example, the a(1) = 1 through a(5) = 22 trees are:
%C A284778   (ooo)  (oooo)  (ooooo)   (oooooo)   (ooooooo)
%C A284778                  ((oo)oo)  ((oo)ooo)  ((oo)oooo)
%C A284778                  (o(oo)o)  ((ooo)oo)  ((ooo)ooo)
%C A284778                  (oo(oo))  (o(oo)oo)  ((oooo)oo)
%C A284778                            (o(ooo)o)  (o(oo)ooo)
%C A284778                            (oo(oo)o)  (o(ooo)oo)
%C A284778                            (oo(ooo))  (o(oooo)o)
%C A284778                            (ooo(oo))  (oo(oo)oo)
%C A284778                                       (oo(ooo)o)
%C A284778                                       (oo(oooo))
%C A284778                                       (ooo(oo)o)
%C A284778                                       (ooo(ooo))
%C A284778                                       (oooo(oo))
%C A284778                                       (((oo)o)oo)
%C A284778                                       ((o(oo))oo)
%C A284778                                       ((oo)(oo)o)
%C A284778                                       ((oo)o(oo))
%C A284778                                       (o((oo)o)o)
%C A284778                                       (o(o(oo))o)
%C A284778                                       (o(oo)(oo))
%C A284778                                       (oo((oo)o))
%C A284778                                       (oo(o(oo)))
%C A284778 (End)
%H A284778 Alois P. Heinz, <a href="/A284778/b284778.txt">Table of n, a(n) for n = 0..2105</a>
%H A284778 Alois P. Heinz, <a href="/A284778/a284778.gif">Animation of a(6)=54 walks</a>
%F A284778 G.f.: (1-2*x-x^2-sqrt(1-4*x+2*x^2+4*x^3-3*x^4))/(2*(x+1)*x^3).
%F A284778 Recursion: see Maple program.
%F A284778 a(n) = A284414(n,n+1) = A284652(n,n+1).
%F A284778 a(n) ~ 3^(n+5/2) / (4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 02 2017
%F A284778 a(n) = Sum_{k=0..floor(n/2)} (k+1)^2/(n-k)*Sum_{i=0..n-1-2*k} C(i,n-1-2*k-i)*C(n-k,i), n>0, a(0)=0. - _Vladimir Kruchinin_, Mar 20 2023
%p A284778 a:= proc(n) option remember; `if`(n<3, (3-n)*n/2,
%p A284778       ((n^2-n+3)*a(n-1)+(5*n-2)*n*a(n-2)+
%p A284778        3*(n-1)*n*a(n-3))/((n+3)*(n-1)))
%p A284778     end:
%p A284778 seq(a(n), n=0..35);
%t A284778 CoefficientList[Series[(1 - 2*x - x^2 - Sqrt[1 - 4*x + 2*x^2 + 4*x^3 - 3*x^4])/(2*(x + 1)*x^3), {x, 0, 50}], x] (* _Indranil Ghosh_, Apr 02 2017 *)
%o A284778 (Maxima)
%o A284778 a(n):=if n=0 then 0 else sum(((k+1)^2*sum(binomial(i,n-1-2*k-i)*binomial(n-k,i),i,0,n-1-2*k))/(n-k),k,0,floor((n)/2)); /* _Vladimir Kruchinin_, Mar 20 2023 */
%Y A284778 First upper diagonal of A284414, A284652.
%Y A284778 CF. A005043, A187306.
%K A284778 nonn,walk
%O A284778 0,4
%A A284778 _Alois P. Heinz_, Apr 02 2017