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Expansion of Product_{k>=1} (1 + x^k)^binomial(2*k,k).
1

%I #10 May 10 2021 06:23:59

%S 1,2,7,32,131,562,2383,10124,42916,181844,769246,3250388,13716377,

%T 57812466,243382957,1023463628,4299199426,18040918456,75632083258,

%U 316774424568,1325591467994,5542462776932,23155074355078,96661979245880,403223735948096,1680858909265768

%N Expansion of Product_{k>=1} (1 + x^k)^binomial(2*k,k).

%F a(n) ~ 2^(2*n - 1/3) * exp(3*n^(1/3)/2^(2/3) - 1 + c) / (sqrt(3*Pi) * n^(5/6)), where c = Sum_{k>=2} (-1)^k * (1 - 1/sqrt(1 - 4^(1-k)))/k = -0.0680700790487788003241709878640131435678833005035785286095068...

%t nmax = 25; CoefficientList[Series[Product[(1 + x^k)^Binomial[2 k, k], {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 25; CoefficientList[Series[Exp[Sum[-(-1)^j * (1/Sqrt[1 - 4*x^j] - 1)/j, {j, 1, nmax}]], {x, 0, nmax}], x]

%Y Cf. A000984, A304787, A344108.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, May 10 2021