A327727 Expansion of Product_{i>=1, j>=0} (1 + x^(i*2^j)) / (1 - x^(i*2^j)).

1, 2, 6, 12, 28, 52, 104, 184, 340, 578, 1004, 1652, 2752, 4404, 7088, 11080, 17362, 26592, 40730, 61284, 92096, 136408, 201608, 294456, 428952, 618658, 889684, 1268624, 1803520, 2545164, 3580784, 5005584, 6976046, 9667164, 13356364, 18360368, 25165732

Offset

0,2

Comments

Convolution of the sequences A000041 and A092119.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Andrew Howroyd)

Formula

G.f.: Product_{k>=1} ((1 + x^k) / (1 - x^k))^A001511(k).

G.f.: Product_{k>=1} 1 / (1 - x^k)^(A001511(k) + 1).

log(a(n)) ~ Pi*sqrt(2*n). - Vaclav Kotesovec, Feb 21 2026

Mathematica

nmax = 36; CoefficientList[Series[Product[1/(1 - x^k)^(IntegerExponent[2 k, 2] + 1), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (IntegerExponent[2 d, 2] + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 36}]

Prog

(PARI) seq(n)={Vec(1/prod(k=1, n, (1 - x^k + O(x*x^n))^(2+valuation(k, 2))))} \\ Andrew Howroyd, Sep 23 2019

Crossrefs

Cf. A000041, A001511, A085058, A092119, A170925.

Sequence in context: A290674 A290419 A159553 * A222970 A112510 A384497

Adjacent sequences: A327724 A327725 A327726 * A327728 A327729 A327730

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 23 2019

Status

approved