A283799 - OEIS

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A283799

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

Recursion: see Maple program.

a(n) = A282869(2n,n).

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)

MAPLE

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);

MATHEMATICA

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];