A320586 Expansion of (1/(1 - x)) * Sum_{k>=1} k*x^k/(x^k + (1 - x)^k).

1, 3, 10, 27, 66, 156, 365, 843, 1909, 4238, 9274, 20136, 43564, 94013, 202155, 432475, 919820, 1945767, 4098852, 8610922, 18061277, 37844128, 79212323, 165565920, 345412341, 719047566, 1493488927, 3095654281, 6405734456, 13238611241, 27336762272, 56416256443, 116376652600

Offset: 1

Author

Ilya Gutkovskiy, Oct 16 2018

Keywords

• nonn

Comments

Binomial transform of A000593.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Vaclav Kotesovec, Plot of a(n)/(n*2^n) for n = 1..10000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000593, A129519, A185003, A320589.

Programs

Magma

m:=50; R:=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

Maple

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

Mathematica

nmax = 33; Rest[CoefficientList[Series[1/(1 - x) Sum[k x^k/(x^k + (1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
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]]
Table[Sum[Binomial[n, k] Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}], {k, n}], {n, 33}]

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

Formula

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.

L.g.f.: Sum_{k>=1} A000593(k)*x^k/(k*(1 - x)^k) = Sum_{n>=1} a(n)*x^n/n.

a(n) = Sum_{k=1..n} binomial(n,k)*A000593(k).

Conjecture: a(n) ~ c * 2^n * n, where c = Pi^2/24 = 0.411233516712... - Vaclav Kotesovec, Jun 26 2019