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A304787
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Expansion of Product_{k>=1} (1 + x^k)^(binomial(2*k,k)/(k+1)).
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3
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1, 1, 2, 7, 20, 67, 222, 758, 2617, 9189, 32554, 116494, 420046, 1525221, 5571065, 20457808, 75476447, 279636977, 1039965746, 3880891892, 14527657602, 54537434161, 205270200229, 774460385687, 2928429307876, 11095878177649, 42122749335654, 160192845018335, 610224764470011
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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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G.f.: Product_{k>=1} (1 + x^k)^A000108(k).
a(n) ~ c * 4^n / n^(3/2), where c = exp[Sum_{k>=1} (-1)^k * (2 - 4^k + 4^k*sqrt(1 - 4^(1-k)))/(2*k) ) / sqrt(Pi) = 1.4863036894111457491052224706533674748514957... - Vaclav Kotesovec, Mar 21 2021
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MATHEMATICA
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nmax = 28; CoefficientList[Series[Product[(1 + x^k)^CatalanNumber[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d CatalanNumber[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]
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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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