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%I #18 Sep 08 2022 08:46:23
%S 1,-1,4,-13,36,-88,197,-421,895,-1946,4346,-9832,22140,-49043,106389,
%T -226213,473366,-980413,2022418,-4179198,8687753,-18201140,38398455,
%U -81343408,172383461,-364158198,764854519,-1595107695,3302884966,-6796646603,13921482698,-28437025029,58034908034
%N Expansion of (1/(1 + x)) * Sum_{k>=1} k*x^k/(x^k + (1 + x)^k).
%C Inverse binomial transform of A000593.
%H G. C. Greubel, <a href="/A320589/b320589.txt">Table of n, a(n) for n = 1..1000</a>
%H Vaclav Kotesovec, <a href="/A320589/a320589.jpg">Plot of |a(n)|/(n*2^n) for n = 1..10000</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f.: (theta_3(x/(1 + x))^4 + theta_2(x/(1 + x))^4 - 1)/(24*(1 + x)), where theta_() is the Jacobi theta function.
%F L.g.f.: Sum_{k>=1} A000593(k)*x^k/(k*(1 + x)^k) = Sum_{n>=1} a(n)*x^n/n.
%F a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*A000593(k).
%F Conjecture: a(n) ~ -(-1)^n * c * 2^n * n, where c = Pi^2/48 = 0.205616758356... - _Vaclav Kotesovec_, Jun 26 2019
%p seq(coeff(series((1/(1+x))*add(k*x^k/(x^k+(1+x)^k),k=1..n),x,n+1), x, n), n = 1 .. 35); # _Muniru A Asiru_, Oct 16 2018
%t nmax = 33; Rest[CoefficientList[Series[1/(1 + x) Sum[k x^k/(x^k + (1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
%t nmax = 33; Rest[CoefficientList[Series[(EllipticTheta[3, 0, x/(1 + x)]^4 + EllipticTheta[2, 0, x/(1 + x)]^4 - 1)/(24 (1 + x)), {x, 0, nmax}], x]]
%t Table[Sum[(-1)^(n - k) Binomial[n, k] Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}], {k, n}], {n, 33}]
%o (PARI) m=50; x='x+O('x^m); Vec((1/(1 + x))*sum(k=1, m+2, k*x^k/(x^k + (1 + x)^k))) \\ _G. C. Greubel_, Oct 29 2018
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1/(1 + x))*(&+[k*x^k/(x^k + (1 + x)^k): k in [1..(m+2)]]) )); // _G. C. Greubel_, Oct 29 2018
%Y Cf. A000593, A320568, A320586, A320591.
%K sign
%O 1,3
%A _Ilya Gutkovskiy_, Oct 16 2018
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