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Expansion of (1/(1 - x)) * Sum_{k>=1} k*x^k/(x^k + (1 - x)^k).
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%I #18 Sep 08 2022 08:46:23

%S 1,3,10,27,66,156,365,843,1909,4238,9274,20136,43564,94013,202155,

%T 432475,919820,1945767,4098852,8610922,18061277,37844128,79212323,

%U 165565920,345412341,719047566,1493488927,3095654281,6405734456,13238611241,27336762272,56416256443,116376652600

%N Expansion of (1/(1 - x)) * Sum_{k>=1} k*x^k/(x^k + (1 - x)^k).

%C Binomial transform of A000593.

%H G. C. Greubel, <a href="/A320586/b320586.txt">Table of n, a(n) for n = 1..1000</a>

%H Vaclav Kotesovec, <a href="/A320586/a320586.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} binomial(n,k)*A000593(k).

%F Conjecture: a(n) ~ c * 2^n * n, where c = Pi^2/24 = 0.411233516712... - _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[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 30 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 30 2018

%Y Cf. A000593, A129519, A185003, A320589.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Oct 16 2018