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%I A298906 #14 Jul 01 2021 11:48:34
%S A298906 1,1,1,4,2,1,77,29,-4289,-14836,283812,1316855,-16548717,-292820579,
%T A298906 911200565,52594983250,100157634380,-3444629077653,7961210574683,
%U A298906 -2170805244559295,-41176659971108705,348776485253486302,35663019455311634058,513993485453689440281
%N A298906 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^(1/k!).
%H A298906 Seiichi Manyama, <a href="/A298906/b298906.txt">Table of n, a(n) for n = 0..450</a>
%F A298906 E.g.f.: exp(Sum_{k>=1} (-1)^(k+1)*(exp(x^k) - 1)/k).
%F A298906 E.g.f.: Product_{k>=1} B(x^k)^((-1)^(k+1)/k), where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers (A000110).
%e A298906 E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 2*x^4/4! + x^5/5! + 77*x^6/6! + ... = (1 + x) * (1 + x^2)^(1/2!) * (1 + x^3)^(1/3!) * (1 + x^4)^(1/4!) * ...
%p A298906 a:=series(exp(add((-1)^(k+1)*(exp(x^k)-1)/k,k=1..100)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # _Paolo P. Lava_, Mar 26 2019
%t A298906 nmax = 23; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) (Exp[x^k] - 1)/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A298906 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1)/(d - 1)!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 23}]
%Y A298906 Cf. A000110, A168243, A209902, A345762.
%K A298906 sign
%O A298906 0,4
%A A298906 _Ilya Gutkovskiy_, Jun 18 2018

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