A320286 Expansion of Product_{k>=1} 1/(1 - x^k - x^(2*k) - x^(3*k)).

1, 1, 3, 6, 13, 24, 51, 93, 184, 343, 654, 1211, 2286, 4217, 7865, 14521, 26912, 49600, 91669, 168800, 311305, 573058, 1055576, 1942437, 3575840, 6578762, 12106121, 22270404, 40972700, 75367724, 138644224, 255020102, 469095029, 862827347, 1587061299, 2919111935, 5369224903

Offset: 0

Author

Ilya Gutkovskiy, Oct 09 2018

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000010, A001935, A082303, A093305, A162891.

Programs

Magma

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/( &*[(1-x^k-x^(2*k)-x^(3*k)): k in [1..m+2]]))); // G. C. Greubel, Oct 24 2018

Maple

seq(coeff(series(mul(((1-x^k-x^(2*k)-x^(3*k)))^(-1),k=1..n),x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Oct 25 2018

Mathematica

nmax = 36; CoefficientList[Series[Product[1/(1 - x^k - x^(2 k) - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 36; CoefficientList[Series[Exp[-Sum[Sum[EulerPhi[j] Log[1 - x^(j k) (1 + x^(j k) + x^(2 j k))]/(j k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]

PARI

m=40; x='x+O('x^m); Vec(1/prod(k=1, m+2, (1-x^k-x^(2*k)-x^(3*k)))) \\ G. C. Greubel, Oct 24 2018

Formula

G.f.: exp(-Sum_{k>=1} Sum_{j>=1} phi(j)*log(1 - x^(j*k)*(1 + x^(j*k) + x^(2*j*k)))/(j*k)), where phi = Euler totient function (A000010).

From Vaclav Kotesovec, Oct 09 2018: (Start)

a(n) ~ s*p / r^(n+1), where

r = A192918 = ((17 + 3*sqrt(33))^(1/3) - 2/(17 + 3*sqrt(33))^(1/3) - 1)/3 = 0.54368901269207636157085597180174798652520329765098393524... is the real root of the equation 1 - r - r^2 - r^3 = 0,

s = (51 + 9*sqrt(33))/(4*(17 + 3*sqrt(33))^(1/3) + (17 + 3*sqrt(33))^(5/3) - 34 - 6*sqrt(33)) = 0.3362281169949410942253629540143324151579260900204592... is the real root of the equation -1 - 2*s + 44*s^3 = 0,

p = Product_{k>=2} 1/(1 - r^k - r^(2*k) - r^(3*k)) = 2.577933056783997593784130068093034525002002622982961271582417329674...

(End)