A318570 Expansion of Product_{k>=1} ((1 - x)^k + x^k)/((1 - x)^k - x^k).

1, 2, 6, 18, 52, 146, 402, 1090, 2916, 7708, 20160, 52236, 134222, 342304, 867024, 2182384, 5461696, 13595918, 33677550, 83036878, 203859820, 498470998, 1214230586, 2947204870, 7129403128, 17191258642, 41328057106, 99067295658, 236822823336, 564650823162, 1342921372126

Offset: 0

Author

Ilya Gutkovskiy, Oct 15 2018

Keywords

• nonn

Comments

First differences of the binomial transform of A015128.

Convolution of A129519 and A218482.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000203, A015128, A129519, A218482, A266497.

Programs

Maple

a:=series(mul(((1-x)^k+x^k)/((1-x)^k-x^k),k=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 30; CoefficientList[Series[Product[((1 - x)^k + x^k)/((1 - x)^k - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[1/EllipticTheta[4, 0, x/(1 - x)], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: 1/theta_4(x/(1 - x)), where theta_4() is the Jacobi theta function.

G.f.: exp(Sum_{k>=1} (sigma(2*k) - sigma(k))*x^k/(k*(1 - x)^k)).

a(n) ~ 2^(n-3) * exp(Pi*sqrt(n/2) + Pi^2/16) / n. - Vaclav Kotesovec, Oct 15 2018