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A283799 Number of dispersed Dyck prefixes of length 2n and height n.

%I A283799 #16 Jan 17 2025 18:38:43
%S A283799 1,2,5,12,36,90,286,728,2380,6120,20349,52668,177100,460460,1560780,
%T A283799 4071600,13884156,36312408,124403620,326023280,1121099408,2942885946,
%U A283799 10150595910,26681566392,92263734836,242799302200,841392966470,2216352204360,7694644696200
%N A283799 Number of dispersed Dyck prefixes of length 2n and height n.
%H A283799 Alois P. Heinz, <a href="/A283799/b283799.txt">Table of n, a(n) for n = 0..1000</a>
%F A283799 Recursion: see Maple program.
%F A283799 a(n) = A282869(2*n, n).
%F A283799 From _Vaclav Kotesovec_, Mar 26 2018: (Start)
%F A283799 Recurrence: 3*n*(3*n + 1)*(3*n + 2)*(3*n^3 - 11*n^2 + 10*n - 3)*a(n) =  - 24*(2*n - 1)*(6*n^3 - 1)*a(n-1) + 64*(n-1)*(2*n - 3)*(2*n - 1)*(3*n^3 - 2*n^2 - 3*n - 1)*a(n-2).
%F A283799 a(n) ~ ((3+2*sqrt(3)) - (-1)^n*(3-2*sqrt(3))) * 2^(4*n + 1) / (sqrt(Pi*n) * 3^(3*n/2 + 2)). (End)
%F A283799 From _Peter Luschny_, Jan 17 2025: (Start)
%F A283799 a(n) = binomial(2*n, n - floor(n/2 + 1/2)) + binomial(2*n, floor(n/2 + 1/2) - 1).
%F A283799 a(n) = A379822(n, (n + 1)/2).  (End)
%p A283799 a:= proc(n) option remember; `if`(n<3, 1+n^2, ((512*(2*n-5))
%p A283799       *(2519*n-1279)*(n-2)*(2*n-3)*a(n-3) +(192*(2*n-3))
%p A283799       *(1710*n^3-443*n^2-4990*n+2483)*a(n-2) -(24*(22671*n^4
%p A283799       -124866*n^3+216436*n^2-129032*n+24526))*a(n-1))
%p A283799        / ((3*n+2)*(27*n+9)*(855*n-1504)*n))
%p A283799     end:
%p A283799 seq(a(n), n=0..30);
%p A283799 a := n -> binomial(2*n, n-iquo(n+1, 2)) + binomial(2*n, iquo(n+1,2)-1):
%p A283799 seq(a(n), n = 0..28);  # _Peter Luschny_, Jan 17 2025
%t A283799 b[x_, y_, m_] := b[x, y, m] = If[x == 0, z^m, If[y > 0, b[x - 1, y - 1, m], 0] + If[y == 0, b[x - 1, y, m], 0] + b[x - 1, y + 1, Max[m, y + 1]]];
%t A283799 a[n_] := Coefficient[b[2n, 0, 0], z, n];
%t A283799 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ in A282869 *)
%Y A283799 Cf. A282869, A283667, A379822.
%K A283799 nonn
%O A283799 0,2
%A A283799 _Alois P. Heinz_, Mar 16 2017