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%I #6 Mar 30 2019 08:39:25
%S 1,5,15,46,159,570,2036,7208,25400,89456,315335,1112286,3923867,
%T 13841052,48818892,172186234,607314043,2142064478,7555322206,
%U 26648517536,93992371863,331521717928,1169314641890,4124305724658,14546896171716,51308559972146,180971133233105,638305788168090
%N a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1)*sigma_2(k+1)*a(n-k), where sigma_2() is the sum of squares of divisors (A001157).
%F G.f.: -x / Sum_{k>=1} k^2*(-x)^k/(1 - (-x)^k).
%F G.f.: 1 / (d/dx) log(Product_{k>=1} (1 - (-x)^k)^k).
%t a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) DivisorSigma[2, k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 27}]
%t nmax = 27; CoefficientList[Series[-x/Sum[k^2 (-x)^k/(1 - (-x)^k), {k, 1, nmax + 1}], {x, 0, nmax}], x]
%t nmax = 27; CoefficientList[Series[1/D[Log[Product[(1 - (-x)^k)^k, {k, 1, nmax + 1}]], x], {x, 0, nmax}], x]
%Y Cf. A001157, A002039, A307241, A320649.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Mar 30 2019