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A289474
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Number of Dyck paths of semilength 4*n and height n.
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2
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1, 1, 127, 26609, 5828185, 1244027317, 258054207727, 52402156308977, 10488803639337560, 2079716852900452250, 409870041643305514072, 80466951297324661639450, 15759844503630649015980996, 3082231209190859509953543989, 602325527248635884295374077954
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) ~ 2^(24*n + 4) / (3^(3*n + 1/2) * 5^(5*n + 7/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 14 2017
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MAPLE
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b:= proc(x, y, k) option remember;
`if`(x=0, 1, `if`(y>0, b(x-1, y-1, k), 0)+
`if`(y < min(x-1, k), b(x-1, y+1, k), 0))
end:
a:= n-> `if`(n=0, 1, b(8*n, 0, n)-b(8*n, 0, n-1)):
seq(a(n), n=0..20);
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MATHEMATICA
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b[x_, y_, k_]:=b[x, y, k]=If[x==0, 1, If[y>0, b[x - 1, y - 1, k], 0] + If[y<Min[x - 1, k], b[x - 1, y + 1, k], 0]]; a[n_]:=a[n]=If[n==0, 1, b[8n, 0, n] - b[8n, 0, n - 1]]; Table[a[n], {n, 0, 20}] (* Indranil Ghosh, Jul 08 2017 *)
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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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