A279226 Expansion of Product_{k>=1} (1 + x^(k^2))^2.

1, 2, 1, 0, 2, 4, 2, 0, 1, 4, 5, 2, 0, 4, 8, 4, 2, 6, 7, 4, 5, 8, 6, 4, 4, 10, 15, 8, 1, 12, 24, 12, 1, 8, 19, 18, 10, 8, 16, 24, 17, 16, 23, 20, 12, 22, 34, 20, 8, 20, 42, 38, 18, 18, 42, 52, 30, 20, 34, 46, 34, 30, 46, 48, 36, 46, 72, 58, 33, 42, 71, 72, 41

Offset: 0

Author

Vaclav Kotesovec, Dec 08 2016

Keywords

• nonn

Links

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

Crossrefs

Cf. A033461, A103265, A279225, A279227.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^(k^2))^2, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 20; poly = ConstantArray[0, nmax^2 + 1]; poly[[1]] = 1; poly[[2]] = 2; poly[[3]] = 1; Do[Do[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, nmax^2, k^2, -1}];, {p, 1, 2}], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Dec 09 2016 *)

Formula

a(n) ~ exp(3 * Pi^(1/3) * ((sqrt(2)-1) * Zeta(3/2))^(2/3) * n^(1/3) / 2) * sqrt(2/3) * ((sqrt(2)-1) * Zeta(3/2) / Pi)^(1/3) / (4*n^(5/6)). - Vaclav Kotesovec, Dec 09 2016