A279360 Expansion of Product_{k>=1} (1+2*x^(k^2)).

1, 2, 0, 0, 2, 4, 0, 0, 0, 2, 4, 0, 0, 4, 8, 0, 2, 4, 0, 0, 4, 8, 0, 0, 0, 6, 12, 0, 0, 12, 24, 0, 0, 0, 4, 8, 2, 4, 8, 16, 4, 12, 8, 0, 0, 12, 24, 0, 0, 10, 28, 16, 4, 12, 24, 32, 8, 16, 4, 8, 0, 12, 32, 16, 2, 32, 56, 0, 4, 16, 24, 16, 0, 4, 36, 56, 0, 16

Offset: 0

Author

Vaclav Kotesovec, Dec 10 2016

Keywords

• nonn

Links

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

Crossrefs

Cf. A001422, A032302, A033461, A279226, A279368, A341040.

Programs

Mathematica

nmax = 200; CoefficientList[Series[Product[(1+2*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]] = 2; poly[[3]] = 0; Do[Do[poly[[j + 1]] += 2*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)) / (3 * 2^(2/3) * Pi^(1/3) * n^(5/6)), where c = -PolyLog(3/2, -2) = 1.28138038315976963883198... . - Vaclav Kotesovec, Dec 12 2016

From Alois P. Heinz, Feb 03 2021: (Start)

a(n) = Sum_{k>=0} 2^k * A341040(n,k).

a(n) = 0 <=> n in { A001422 }. (End)