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A289480
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Number of Dyck paths of semilength 10*n and height n.
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2
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1, 1, 524287, 956185155129, 2011805242484811913, 3913893675608035491579363, 6753921048102794214403632812402, 10404372657815158859307324171401493273, 14572291057533118353907127088834174993619633, 18906515358804836479733610566557899759396278209535
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OFFSET
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0,3
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COMMENTS
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In general, column k>1 of A289481 is asymptotic to 2^(2*k*n + 3) * k^(2*k*n + 1/2) / ((k-1)^((k-1)*n + 1/2) * (k+1)^((k+1)*n + 7/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 14 2017
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LINKS
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FORMULA
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a(n) ~ 2^(40*n + 7/2) * 5^(20*n + 1/2) / (3^(18*n + 1) * 11^(11*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(20*n, 0, n)-b(20*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[20n, 0, n] - b[20n, 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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