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A178854
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Asymptotic value of odd Catalan numbers mod 2^n.
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1
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0, 1, 1, 5, 13, 29, 29, 93, 221, 221, 733, 1757, 3805, 7901, 7901, 24285, 57053, 122589, 122589, 384733, 384733, 384733, 2481885, 2481885, 10870493, 10870493, 10870493, 10870493, 145088221
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OFFSET
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0,4
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COMMENTS
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For every n, the odd Catalan numbers C(2^m-1) are eventually constant mod 2^n (namely for m >= n-1): then a(n) is the asymptotic value of the remainder.
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LINKS
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FORMULA
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a(n) = remainder(Catalan(2^m-1), 2^n) for any m >= n-1.
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EXAMPLE
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The odd Catalan numbers mod 2^6=64 are 1,5,45,61,29,29,29, so a(6)=29.
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MAPLE
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A000108 := proc(n) binomial(2*n, n)/(n+1) ; end proc:
A178854 := proc(n) if n = 0 then 0; else modp(A038003(n-1), 2^n) ; end if; end proc:
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MATHEMATICA
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(* first do *) Needs["DiscreteMath`CombinatorialFunctions`"] (* then *) f[n_] := Mod[ CatalanNumber[2^n - 1], 2^n]; Array[f, 25, 0] (* Robert G. Wilson v, Jun 28 2010 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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