A294749 Expansion of Product_{k>=1} (1 + x^(2*k - 1))^(k^2).

1, 1, 0, 4, 4, 9, 15, 22, 52, 65, 129, 190, 335, 534, 814, 1399, 2074, 3462, 5135, 8303, 12658, 19562, 30182, 45542, 70620, 105034, 161223, 239532, 362929, 539252, 805320, 1197589, 1769483, 2624604, 3847755, 5681787, 8291848, 12165978, 17696362, 25796820

Offset: 0

Author

Vaclav Kotesovec, Nov 08 2017

Keywords

• nonn

Comments

In general, if g.f. = Product_{k>=1} (1 + x^(2*k-1))^(c2*k^2 + c1*k + c0) and c2>0, then a(n) ~ exp(Pi*sqrt(2)/3 * (7*c2/15)^(1/4) * n^(3/4) + 3*(c1+c2) * Zeta(3) / (2*Pi^2) * sqrt(15*n/(7*c2)) + (Pi*(4*c0 + 2*c1 + c2) * (15/(7*c2))^(1/4) / (24*sqrt(2)) - 9*(c1+c2)^2 * Zeta(3)^2 * (15/(7*c2))^(5/4) / (2^(3/2) * Pi^5)) * n^(1/4) + 2025*(c1+c2)^3 * Zeta(3)^3 / (49 * c2^2 * Pi^8) - 15*(c1+c2) * (4*c0 + 2*c1 + c2) * Zeta(3) / (112 * c2 * Pi^2)) * (7/15)^(1/8) * 2^((c1+c2)/24 - 9/4) * c2^(1/8) / n^(5/8).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A027998, A263140, A294750, A294755.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1+x^(2*k-1))^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi/3 * (7/15)^(1/4) * sqrt(2) * n^(3/4) + 3*Zeta(3) * sqrt(15*n/7) / (2*Pi^2) + (Pi * (15/7)^(1/4) / (24*sqrt(2)) - 9*Zeta(3)^2 * (15/7)^(5/4) / (2^(3/2) * Pi^5)) * n^(1/4) + 2025*Zeta(3)^3 / (49*Pi^8) - 15*Zeta(3) / (112*Pi^2)) * (7/15)^(1/8) / (2^(53/24) * n^(5/8)).