A279368 Expansion of Product_{k>=1} (1+3*x^(k^2)).

1, 3, 0, 0, 3, 9, 0, 0, 0, 3, 9, 0, 0, 9, 27, 0, 3, 9, 0, 0, 9, 27, 0, 0, 0, 12, 36, 0, 0, 36, 108, 0, 0, 0, 9, 27, 3, 9, 27, 81, 9, 36, 27, 0, 0, 36, 108, 0, 0, 30, 117, 81, 9, 36, 108, 243, 27, 81, 9, 27, 0, 36, 135, 81, 3, 126, 351, 0, 9, 54, 108, 81, 0

Offset: 0

Author

Vaclav Kotesovec, Dec 10 2016

Keywords

• nonn

Comments

In general, if m > 0 and g.f. = Product_{k>=1} (1 + m*x^(k^2)), then a(n) ~ exp(3 * 2^(-4/3) * Pi^(1/3) * c^(2/3) * n^(1/3)) * c^(1/3) / (2^(2/3) * Pi^(1/3) * sqrt(3*(m+1)) * n^(5/6)), where c = -PolyLog(3/2, -m). - Vaclav Kotesovec, Dec 12 2016

Links

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

Crossrefs

Cf. A032308, A033461, A279360.

Programs

Mathematica

nmax = 200; CoefficientList[Series[Product[(1+3*x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 200; nn = Floor[Sqrt[nmax]]+1; poly = ConstantArray[0, nn^2 + 1]; poly[[1]] = 1; poly[[2]] = 3; poly[[3]] = 0; Do[Do[poly[[j + 1]] += 3*poly[[j - k^2 + 1]], {j, nn^2, k^2, -1}];, {k, 2, nn}]; Take[poly, nmax+1]

Formula

a(n) ~ c^(1/3) * exp(3 * 2^(-4/3) * c^(2/3) * Pi^(1/3) * n^(1/3)) / (2^(5/3) * sqrt(3) * Pi^(1/3) * n^(5/6)), where c = -PolyLog(3/2, -3) = 1.679089730504828... . - Vaclav Kotesovec, Dec 12 2016