A327744 Expansion of Product_{i>=1, j>=1} 1 / (1 - x^(i*j*(j + 1)/2)).

1, 1, 2, 4, 6, 9, 17, 23, 35, 54, 77, 108, 163, 221, 309, 436, 593, 800, 1109, 1470, 1968, 2642, 3482, 4566, 6052, 7848, 10204, 13276, 17092, 21924, 28245, 35949, 45762, 58231, 73609, 92789, 117140, 146799, 183826, 229995, 286483, 356040, 442566, 547489

Offset: 0

Author

Ilya Gutkovskiy, Sep 23 2019

Keywords

• nonn

Comments

Euler transform of A007862.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A004101, A007862, A327745.

Programs

Mathematica

nmax = 43; CoefficientList[Series[Product[1/(1 - x^k)^Length[Select[Divisors[k], IntegerQ[Sqrt[8 # + 1]] &]], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Length[Select[Divisors[d], IntegerQ[Sqrt[8 # + 1]] &]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]
nmax = 50; CoefficientList[Series[Product[1/QPochhammer[x^(k*(k + 1)/2)], {k, 1, Sqrt[2*nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 24 2019 *)

Formula

G.f.: Product_{k>=1} 1 / (1 - x^k)^A007862(k).