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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. 10
 0, 1, 1, 4, 8, 22, 54, 142, 370, 983, 2627, 7086, 19238, 52561, 144377, 398518, 1104794, 3074809, 8588093, 24064642, 67630898, 190584766, 538412426, 1524554956, 4326119748, 12300296227, 35037658099, 99977847308, 285741659312, 817901027070, 2344475178110 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS From Gus Wiseman, Nov 15 2022: (Start) 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: (ooo) (oooo) (ooooo) (oooooo) (ooooooo) ((oo)oo) ((oo)ooo) ((oo)oooo) (o(oo)o) ((ooo)oo) ((ooo)ooo) (oo(oo)) (o(oo)oo) ((oooo)oo) (o(ooo)o) (o(oo)ooo) (oo(oo)o) (o(ooo)oo) (oo(ooo)) (o(oooo)o) (ooo(oo)) (oo(oo)oo) (oo(ooo)o) (oo(oooo)) (ooo(oo)o) (ooo(ooo)) (oooo(oo)) (((oo)o)oo) ((o(oo))oo) ((oo)(oo)o) ((oo)o(oo)) (o((oo)o)o) (o(o(oo))o) (o(oo)(oo)) (oo((oo)o)) (oo(o(oo))) (End) LINKS Alois P. Heinz, Table of n, a(n) for n = 0..2105 Alois P. Heinz, Animation of a(6)=54 walks FORMULA 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). Recursion: see Maple program. a(n) = A284414(n,n+1) = A284652(n,n+1). a(n) ~ 3^(n+5/2) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 02 2017 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 MAPLE a:= proc(n) option remember; `if`(n<3, (3-n)*n/2, ((n^2-n+3)*a(n-1)+(5*n-2)*n*a(n-2)+ 3*(n-1)*n*a(n-3))/((n+3)*(n-1))) end: seq(a(n), n=0..35); MATHEMATICA 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 *) PROG (Maxima) 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 */ CROSSREFS First upper diagonal of A284414, A284652. CF. A005043, A187306. Sequence in context: A297339 A290138 A266922 * A057583 A129788 A170938 Adjacent sequences: A284775 A284776 A284777 * A284779 A284780 A284781 KEYWORD nonn,walk AUTHOR Alois P. Heinz, Apr 02 2017 STATUS approved

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Last modified July 18 12:17 EDT 2024. Contains 374378 sequences. (Running on oeis4.)