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%I #15 Oct 25 2018 05:14:59
%S 1,1,-2,5,-14,40,-111,293,-731,1726,-3882,8408,-17796,37423,-79337,
%T 170917,-373812,823585,-1809844,3934750,-8424747,17749392,-36874749,
%U 75862016,-155359339,318331170,-655146929,1356952623,-2828151136,5920984735,-12420140296,26036563525
%N a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*sigma(k).
%C Inverse binomial transform of A000203.
%H Vaclav Kotesovec, <a href="/A320568/b320568.txt">Table of n, a(n) for n = 1..3000</a>
%H Vaclav Kotesovec, <a href="/A320568/a320568.jpg">Graph abs(a(n)) / (n*2^n)</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f.: (1/(1 + x))*Sum_{k>=1} k*x^k/((1 + x)^k - x^k).
%F L.g.f.: Sum_{k>=1} sigma(k)*x^k/(k*(1 + x)^k) = Sum_{n>=1} a(n)*x^n/n.
%F Conjecture: a(n) ~ (-1)^n * Pi^2/48 * n * 2^n. - _Vaclav Kotesovec_, Oct 16 2018
%p seq(add((-1)^(n-k)*binomial(n,k)*sigma(k),k=1..n),n=1..32); # _Paolo P. Lava_, Oct 25 2018
%t Table[Sum[(-1)^(n - k) Binomial[n, k] DivisorSigma[1, k], {k, n}], {n, 32}]
%t nmax = 32; Rest[CoefficientList[Series[1/(1 + x) Sum[k x^k/((1 + x)^k - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
%o (GAP) Flat(List([1..10],n->Sum([1..n],k->(-1)^(n-k)*Binomial(n,k)*Sigma(k)))); # _Muniru A Asiru_, Oct 15 2018
%Y Cf. A000203, A038200, A185003.
%K sign
%O 1,3
%A _Ilya Gutkovskiy_, Oct 15 2018