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%I #4 May 27 2020 20:16:37
%S 1,1,-3,-2,9,0,-32,18,108,-118,-333,576,911,-2466,-2040,9702,2529,
%T -35622,8254,122436,-88275,-391882,501660,1148334,-2331810,-2949282,
%U 9689949,5791930,-37155906,-2645148,133051344,-54698868,-445531893,408566282,1383325848,-2115234972
%N G.f.: x / (Sum_{k>=1} x^k / (1 + x^k)^2).
%F G.f.: x / (Sum_{k>=1} (-1)^(k+1) * k * x^k / (1 - x^k)).
%F G.f.: 1 / log(g(x))', where g(x) = Sum_{k>=0} x^(k*(k + 1)/2) is the g.f. for A010054.
%F G.f.: 1 / (Sum_{k>=0} A002129(k+1) * x^k).
%F a(0) = 1; a(n) = -Sum_{k=1..n} A002129(k+1) * a(n-k).
%t nmax = 35; CoefficientList[Series[x/Sum[x^k/(1 + x^k)^2, {k, 1, nmax + 1}], {x, 0, nmax}], x]
%t nmax = 35; CoefficientList[Series[1/D[Log[Sum[x^(k (k + 1)/2), {k, 0, nmax}]], x], {x, 0, nmax}], x]
%t a[0] = 1; a[n_] := a[n] = -Sum[DivisorSum[k + 1, (-1)^(# + 1) # &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 35}]
%Y Cf. A002039, A002129, A010054, A180305, A335227.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, May 27 2020
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