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A283667
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Number of Motzkin prefixes of length 2n and height n.
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3
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1, 3, 13, 64, 334, 1802, 9933, 55575, 314362, 1793126, 10295625, 59430043, 344559826, 2005026610, 11703965955, 68503652100, 401892122682, 2362629703214, 13914547415998, 82081163986020, 484893156220356, 2868234297838092, 16986185485228431, 100703275233924096
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OFFSET
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0,2
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LINKS
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FORMULA
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Recursion: see Maple program.
a(n) ~ sqrt(769 + 2762/sqrt(13)) * (70 + 2*13^(3/2))^n / (3^(3*n+3)*sqrt(3*Pi*n)). - Vaclav Kotesovec, Mar 13 2017
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MAPLE
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a:= proc(n) option remember; `if`(n<2, 1+2*n, ((4*n-2)*(455*n^6-
1155*n^5-2776*n^4+1047*n^3+1493*n^2-72*n-72)*a(n-1)+36*
(n-1)*(2*n-1)*(2*n-3)*(13*n^4-7*n^3-64*n^2-44*n-6)*a(n-2))/
((9*n+6)*(3*n+1)*(n+1)*(13*n^4-59*n^3+35*n^2+11*n-6)))
end:
seq(a(n), n=0..30);
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MATHEMATICA
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b[x_, y_, m_] := b[x, y, m] = If[x == 0, z^m, b[x - 1, y, m] + If[y > 0, b[x - 1, y - 1, m], 0] + b[x - 1, y + 1, Max[m, y + 1]]];
a[n_] := SeriesCoefficient[b[2n, 0, 0], {z, 0, n}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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