A307057 Expansion of 1/(2 - Product_{k>=2} 1/(1 - x^k)).

1, 0, 1, 1, 3, 4, 10, 15, 33, 55, 111, 195, 380, 684, 1306, 2389, 4507, 8313, 15591, 28881, 53991, 100257, 187086, 347860, 648512, 1206656, 2248399, 4185087, 7796011, 14514195, 27033073, 50334299, 93741325, 174552379, 325067573, 605316388, 1127249250, 2099115548, 3909023438, 7279285948

Offset: 0

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

• nonn

Comments

Invert transform of A002865.

Links

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

Crossrefs

Cf. A002865, A055887, A304969, A307058, A307059, A307060, A307062, A307063.

Programs

Magma

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - 1/(&*[1 - x^j: j in [2..m+2]])) )); // G. C. Greubel, Jan 24 2024

Mathematica

nmax = 50; CoefficientList[Series[1/(2 - Product[1/(1 - x^k), {k, 2, nmax}]), {x, 0, nmax}], x]
a[0]= 1; a[n_]:= a[n]= Sum[(PartitionsP[k] -PartitionsP[k-1]) a[n-k], {k,n}];
Table[a[n], {n,0,50}]
CoefficientList[Series[1/(2 -(1-x)/QPochhammer[x]), {x,0,80}], x] (* G. C. Greubel, Jan 24 2024 *)

SageMath

m=80;
def f(x): return 1/( 2 - (1-x)/product(1 - x^j for j in range(1,m+3)) )
def A307057_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307057_list(m) # G. C. Greubel, Jan 24 2024

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A002865(k)*a(n-k).

a(n) ~ c / r^n, where r = 0.53700045638650021831634004949965496126950171484122... is the root of the equation 1 - r = 2*QPochhammer[r] and c = 0.2143395760756683581919851351414497181589685708674442097498294834747517926...

From G. C. Greubel, Jan 24 2024: (Start)

G.f.: 1/( 2 - (1-x)/QPochhammer(x) ).

G.f.: 1/( 2 - x^(1/24)*(1-x)/eta(x) ), where eta(x) is the Dedekind eta function. (End)