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Expansion of e.g.f. Product_{k>=1} 1 / (2 - exp(x^k)).
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%I #10 Dec 19 2024 06:17:28

%S 1,1,5,25,195,1521,16713,179425,2432139,33902149,546239793,9158893173,

%T 173742256251,3402217292137,73413011744985,1653326843775193,

%U 40118677865954475,1014971456865241197,27429061245764539521,770776923753566642365,22928146838491708702395

%N Expansion of e.g.f. Product_{k>=1} 1 / (2 - exp(x^k)).

%F E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of Fubini numbers (A000670).

%F E.g.f.: Product_{j>=1} 1 / (1 - Sum_{i>=1} x^(i*j) / i!).

%F E.g.f.: exp(Sum_{k>=1} Sum_{d|k} (exp(x^(k/d)) - 1)^d / d).

%F a(n) ~ n! * c / (2 * (log(2))^(n+1)), where c = Product_{k>=2} 1/(2 - exp(log(2)^k)) = 10.38787833857244631... - _Vaclav Kotesovec_, Dec 11 2019

%t nmax = 20; CoefficientList[Series[Product[1/(2 - Exp[x^k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 20; CoefficientList[Series[Exp[Sum[Sum[(Exp[x^(k/d)] - 1)^d/d, {d, Divisors[k]}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

%Y Cf. A000262, A000670, A209903.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Dec 05 2019