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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
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.
Sequence in context: A297339 A290138 A266922 * A057583 A129788 A170938
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.)