A373295 Expansion of 1/Product_{k>=1} (1 - x^k)^(valuation(k,4) + 1).

1, 1, 2, 3, 6, 8, 13, 18, 29, 39, 57, 77, 112, 148, 205, 271, 373, 485, 649, 841, 1116, 1431, 1865, 2379, 3080, 3896, 4979, 6268, 7961, 9953, 12524, 15585, 19505, 24135, 29984, 36943, 45678, 56007, 68841, 84080, 102912, 125164, 152449, 184756, 224184, 270691, 327094, 393675

Offset

0,3

Links

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

Formula

G.f.: A(x) = 1/Product_{i>=1, j>=0} (1 - x^(i * 4^j)).

Let A(x) be the g.f. of this sequence, and P(x) be the g.f. of A000041, then P(x) = A(x)/A(x^4).

log(a(n)) ~ 2*Pi*sqrt(2*n)/3. - Vaclav Kotesovec, Feb 21 2026

Mathematica

nmax = 60; CoefficientList[1/Series[Product[(1 - x^k)^(IntegerExponent[k, 4] + 1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 21 2026 *)

Prog

(PARI) my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^k)^(valuation(k, 4)+1)))

Crossrefs

Cf. A092119, A173241, A373296, A373297, A373298.

Cf. A000041, A115362, A174065.

Sequence in context: A024788 A285472 A318027 * A004101 A003405 A153918

Adjacent sequences: A373292 A373293 A373294 * A373296 A373297 A373298

Keyword

nonn

Author

Seiichi Manyama, May 31 2024

Status

approved