A373220 Expansion of Product_{i>=1, j>=0} (1 + x^(i * 6^j)).

1, 1, 1, 2, 2, 3, 5, 6, 7, 10, 12, 15, 20, 24, 29, 37, 44, 53, 67, 79, 94, 115, 135, 160, 193, 226, 265, 315, 367, 428, 505, 585, 678, 792, 913, 1054, 1225, 1406, 1614, 1862, 2129, 2436, 2797, 3187, 3630, 4147, 4709, 5347, 6084, 6887, 7793, 8832, 9968, 11247, 12706, 14301, 16089, 18116, 20337

Offset

0,4

Links

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

Vaclav Kotesovec, Plot of a(n) / (exp(Pi*sqrt(2*n/5)) / n^(3/4 + log(2)/(4*log(6)))) for n = 1..40000

Formula

G.f.: Product_{k>=1} (1 + x^k)^A373216(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^6).

a(n) ~ 6^(1/4) * Pi^(log(2)/(2*log(6))) * exp(Pi*sqrt(2*n/5)) / (2^(3/2 + (3*log(2) + 2*log(Pi))/(4*log(6))) * 15^(1/4 + log(2)/(4*log(6))) * n^(3/4 + log(2)/(4*log(6)))). - Vaclav Kotesovec, Mar 07 2026

Mathematica

nmax = 60; CoefficientList[Series[Product[(1 + x^k)^(IntegerExponent[k, 6] + 1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 20 2026 *)

Prog

(PARI) my(N=60, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(valuation(k, 6)+1)))

Crossrefs

Cf. A000041, A327726, A174065, A373219, A373221, A393565.

Cf. A000009, A373216.

Sequence in context: A006065 A218933 A266746 * A096981 A281966 A276431

Adjacent sequences: A373217 A373218 A373219 * A373221 A373222 A373223

Keyword

nonn

Author

Seiichi Manyama, May 28 2024

Status

approved