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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]]];
a[n_] := Coefficient[b[2n, 0, 0], z, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz in A282869 *)
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From Vaclav Kotesovec, Mar 26 2018: (Start)
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).
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)
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Alois P. Heinz, <a href="/A283799/b283799.txt">Table of n, a(n) for n = 0..1000</a>
allocated for Alois P. Heinz
Number of dispersed Dyck prefixes of length 2n and height n.
1, 2, 5, 12, 36, 90, 286, 728, 2380, 6120, 20349, 52668, 177100, 460460, 1560780, 4071600, 13884156, 36312408, 124403620, 326023280, 1121099408, 2942885946, 10150595910, 26681566392, 92263734836, 242799302200, 841392966470, 2216352204360, 7694644696200
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Recursion: see Maple program.
a(n) = A282869(2n,n).
a:= proc(n) option remember; `if`(n<3, 1+n^2, ((512*(2*n-5))
*(2519*n-1279)*(n-2)*(2*n-3)*a(n-3) +(192*(2*n-3))
*(1710*n^3-443*n^2-4990*n+2483)*a(n-2) -(24*(22671*n^4
-124866*n^3+216436*n^2-129032*n+24526))*a(n-1))
/ ((3*n+2)*(27*n+9)*(855*n-1504)*n))
end:
seq(a(n), n=0..30);
Cf. A282869.
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Alois P. Heinz, Mar 16 2017
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allocated for Alois P. Heinz
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