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A305256
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Expansion of exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*sqrt(1 - 4*x^k))).
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1
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1, 1, 2, 8, 27, 103, 389, 1497, 5786, 22556, 88230, 346576, 1365119, 5390585, 21327913, 84527939, 335477433, 1333079925, 5302763618, 21112688376, 84125853415, 335443149005, 1338370995240, 5342843332758, 21339341267983, 85266832981905, 340840044333836, 1362936812554758
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OFFSET
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0,3
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COMMENTS
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Weigh transform of the central binomial coefficients 1, 2, 6, 20, 70, ... (A000984).
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LINKS
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FORMULA
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G.f.: Product_{k>=1} (1 + x^k)^binomial(2*k-2,k-1).
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
binomial(binomial(2*i-2, i-1), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
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MATHEMATICA
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nmax = 27; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^k/(k Sqrt[1 - 4 x^k]), {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[Product[(1 + x^k)^Binomial[2 k - 2, k - 1], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Binomial[2 d - 2, d - 1], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]
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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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