A296238 Expansion of Product_{k>0} (1 + x^(k*(3*k+1)/2)).

1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 2, 0, 2, 1, 1, 1, 0

Offset

0,58

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Formula

a(n) ~ zeta(3/2)^(1/3) * (sqrt(2) - 1)^(1/3) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * (sqrt(2) - 1)^(2/3) * n^(1/3) / (2^(5/3)*d^(1/3))) / (2^(4/3 + b/(2*d)) * sqrt(3) * d^(1/6) * Pi^(1/3) * n^(5/6)) * (1 - ((sqrt(2) - 1)^(4/3) * b^2 * Pi^(2/3) * zeta(1/2) * zeta(3/2)^(1/3) / (2^(23/6) * d^(5/3)) + 5*d^(1/3) / (9 * (2*Pi)^(1/3) * (sqrt(2) - 1)^(2/3) * zeta(3/2)^(2/3))) / n^(1/3)), where d = 3/2, b = 1/2. - Vaclav Kotesovec, Mar 11 2026

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^(k*(3*k+1)/2)), {k, 1, Floor[Sqrt[1 + 24*nmax]/6]+1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 10 2017, updated Mar 11 2026 *)

Prog

(PARI) N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+x^(k*(3*k+1)/2))))

Crossrefs

Cf. A218380, A290942, A296237.

Sequence in context: A080843 A287112 A386537 * A309475 A221314 A326642

Adjacent sequences: A296235 A296236 A296237 * A296239 A296240 A296241

Keyword

nonn

Author

Seiichi Manyama, Dec 09 2017

Status

approved