A384426 Expansion of Sum_{k>=1} x^k * Product_{j=k..2*k} (1 + x^j).

0, 1, 2, 2, 3, 2, 3, 3, 4, 4, 4, 5, 6, 5, 6, 7, 8, 8, 9, 9, 10, 12, 12, 13, 14, 14, 16, 18, 19, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 38, 40, 43, 46, 48, 51, 54, 56, 60, 64, 67, 72, 77, 80, 84, 88, 92, 98, 105, 110, 116, 122, 128, 134, 142, 148, 155, 164, 172

Offset

0,3

Comments

Number of partitions p of n with max(p) <= 2*min(p), where the multiplicity of min(p) is 1 or 2 and all other parts are distinct. For example: a(8) = 4 counts these partitions: 8, 53, 44, 422. - Seiichi Manyama, Mar 10 2026

Links

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

Formula

a(n) ~ c * exp(r*sqrt(n)) / sqrt(n), where r = 0.926140105877... = 2*sqrt((3/2)*log(z)^2 - polylog(2, 1-z) + polylog(2, 1-z^2)), where z = (-1 + (44 - 3*sqrt(177))^(1/3) + (44 + 3*sqrt(177))^(1/3))/6 = 0.82948354095849703967... is the real root of the equation z^3*(1 - z)/(1 - z^2)^2 = 1 and c = 0.6975701...

Mathematica

nmax = 100; CoefficientList[Series[Sum[x^k * Product[1 + x^j, {j, k, 2*k}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 + x^(2*k - 1))*(1 + x^(2*k))/(1 + x^k)]; p = Normal[p + O[x]^nmax]; s += p*(1 + x^k)*x^k; , {k, 1, nmax}]; Take[CoefficientList[s, x], nmax + 1]

Crossrefs

Cf. A237824, A207642, A241061.

Sequence in context: A286716 A029213 A029209 * A282630 A108309 A383259

Adjacent sequences: A384423 A384424 A384425 * A384427 A384428 A384429

Keyword

nonn

Author

Vaclav Kotesovec, Jun 14 2025

Status

approved