This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A300853 #20 Dec 18 2023 01:43:41
%S A300853 1,-1,1,3,1,-1,1,-5,10,-1,1,3,1,-1,1,11,1,-10,1,3,1,-1,1,-5,26,-1,10,
%T A300853 3,1,-1,1,-21,1,-1,1,30,1,-1,1,-5,1,-1,1,3,10,-1,1,11,50,-26,1,3,1,
%U A300853 -10,1,-5,1,-1,1,3,1,-1,10,43,1,-1,1,3,1,-1,1,-50,1,-1,26,3,1,-1,1,11,91,-1,1,3,1
%N A300853 L.g.f.: log(Product_{k>=1} (1 + x^(k^2))) = Sum_{n>=1} a(n)*x^n/n.
%H A300853 Andrew Howroyd, <a href="/A300853/b300853.txt">Table of n, a(n) for n = 1..1000</a>
%F A300853 G.f.: Sum_{k>=1} k^2*x^(k^2)/(1 + x^(k^2)).
%F A300853 a(n) = 1 if n is an odd squarefree (A056911).
%F A300853 a(n) = -1 if n is an even squarefree (A039956).
%F A300853 a(n) = Sum_{d^2|n} (-1)^(n/d^2 + 1) * d^2. - _Andrew Howroyd_, Jul 20 2018
%F A300853 Multiplicative with a(2^e) = (1 - (-2)^(e + 1))/3, and a(p^e) = (p^(2*floor(e/2 + 1)) - 1)/(p^2 - 1) for an odd prime p. - _Amiram Eldar_, Oct 25 2020
%F A300853 From _Amiram Eldar_, Dec 18 2023: (Start)
%F A300853 Dirichlet g.f.: zeta(s) * zeta(2*s-2) * (1 - 1/2^(s-1)).
%F A300853 Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (1 - 1/sqrt(2)) * zeta(3/2)/3 = A268682 * A078434 / 3 = 0.255049... . (End)
%e A300853 L.g.f.: L(x) = x - x^2/2 + x^3/3 + 3*x^4/4 + x^5/5 - x^6/6 + x^7/7 - 5*x^8/8 + 10*x^9/9 - x^10/10 + ...
%e A300853 exp(L(x)) = 1 + x + x^4 + x^5 + x^9 + x^10 + x^13 + x^14 + ... + A033461(n)*x^n + ...
%t A300853 nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^k^2), {k, 1, Floor[nmax^(1/2) + 1]}]], {x, 0, nmax}],x] Range[0, nmax]]
%t A300853 nmax = 85; Rest[CoefficientList[Series[Sum[k^2 x^k^2/(1 + x^k^2), {k, 1,Floor[nmax^(1/2) + 1]}], {x, 0, nmax}], x]]
%t A300853 Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[#^(1/2)] &], {n, 85}]
%t A300853 f[p_, e_] := If[p == 2, (1 - (-2)^(e + 1))/3, (p^(2*Floor[e/2 + 1]) - 1)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 25 2020 *)
%o A300853 (PARI) seq(n)={Vec(deriv(log(prod(k=1, n, (1 + x^(k^2) + O(x*x^n))))))} \\ _Andrew Howroyd_, Jul 20 2018
%o A300853 (PARI) a(n)={sumdiv(n, d, if(n%d^2, 0, (-1)^(n/d^2 + 1) * d^2))} \\ _Andrew Howroyd_, Jul 20 2018
%Y A300853 Cf. A000290, A033461, A035316, A039956, A056911, A304876, A304906.
%Y A300853 Cf. A078434, A268682.
%K A300853 sign,mult,easy
%O A300853 1,4
%A A300853 _Ilya Gutkovskiy_, Mar 13 2018
%E A300853 Keyword:mult added by _Andrew Howroyd_, Jul 20 2018