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%I #8 Jun 07 2018 22:01:32
%S 1,-1,2,0,-6,60,-540,3780,12600,-199080,3074400,-45738000,511434000,
%T -5621616000,55394539200,960323364000,-24001273296000,498178528848000,
%U -9994137465312000,156104172544320000,-2076607873660320000,18061446353670720000,206725394268993600000
%N Expansion of e.g.f. log(1 + Sum_{k>=1} x^(k^2)/k^2).
%H Robert Israel, <a href="/A305620/b305620.txt">Table of n, a(n) for n = 1..450</a>
%e E.g.f.: A(x) = x - x^2/2! + 2*x^3/3! - 6*x^5/5! + 60*x^6/6! - 540*x^7/7! + ...
%e exp(A(x)) = 1 + x + x^4/4 + x^9/9 + x^16/16 + ... + x^A000290(k)/A000290(k) + ...
%e exp(exp(A(x))-1) = 1 + x + x^2/2! + x^3/3! + 7*x^4/4! + 31*x^5/5! + ... + A205801(k)*x^k/k! + ... = Product_{j>=1} 1/(1 - x^j)^(A008836(j)/j).
%p N:= 50: # for a(1)..a(N)
%p g:= log(1 + add(x^(k^2)/k^2,k=1..floor(sqrt(N)))):
%p S:= series(g,x,N+1):
%p seq(coeff(S,x,n)*n!,n=1..N); # _Robert Israel_, Jun 07 2018
%t nmax = 23; Rest[CoefficientList[Series[Log[1 + Sum[x^k^2/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
%t nmax = 23; Rest[CoefficientList[Series[Log[1 + Log[Product[1/(1 - x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}]]], {x, 0, nmax}], x] Range[0, nmax]!]
%t a[n_] := a[n] = Boole[IntegerQ[n^(1/2)]] (n - 1)! - Sum[k Binomial[n, k] Boole[IntegerQ[(n - k)^(1/2)]] (n - k - 1)! a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 23}]
%Y Cf. A000290, A008836, A162552, A205801, A205804.
%K sign
%O 1,3
%A _Ilya Gutkovskiy_, Jun 06 2018
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