A327726 Expansion of Product_{i>=1, j>=0} (1 + x^(i * 3^j)).

1, 1, 1, 3, 3, 4, 7, 8, 10, 17, 20, 24, 36, 42, 52, 72, 85, 103, 139, 164, 197, 256, 301, 361, 456, 536, 637, 794, 930, 1098, 1347, 1571, 1848, 2235, 2600, 3042, 3646, 4228, 4922, 5845, 6754, 7835, 9229, 10633, 12284, 14382, 16519, 19013, 22127, 25339, 29073

Offset

0,4

Comments

Weigh transform of A051064.

Links

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

Vaclav Kotesovec, Plot of a(n) / (exp(Pi*sqrt(n/2)) / n^(3/4 + log(2)/(4*log(3)))) for n = 1..100000

Formula

G.f.: Product_{k>=1} (1 + x^k)^A051064(k).

Let A(x) be the g.f. of this sequence, and B(x) be the g.f. of A000009, then B(x) = A(x)/A(x^3). - Seiichi Manyama, May 31 2024

a(n) ~ Pi^(log(2)/(2*log(3))) * exp(Pi*sqrt(n/2)) / (2^(2 + (2*log(2) + log(Pi))/(2*log(3))) * n^(3/4 + log(2)/(4*log(3)))). - Vaclav Kotesovec, Mar 07 2026

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + x^k)^IntegerExponent[3 k, 3], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d IntegerExponent[3 d, 3], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]

Prog

(PARI) seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(1+valuation(k, 3))))} \\ Andrew Howroyd, Sep 23 2019

Crossrefs

Cf. A000041, A174065, A373219, A373220, A373221, A393565.

Cf. A000009, A051064, A143374.

Sequence in context: A343396 A091973 A328798 * A050066 A050064 A137894

Adjacent sequences: A327723 A327724 A327725 * A327727 A327728 A327729

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 23 2019

Status

approved