A296237 Expansion of Product_{k>0} 1/(1 - x^(k*(3*k+1)/2)).

1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 3, 5, 4, 6, 4, 6, 5, 6, 6, 7, 7, 7, 7, 9, 8, 10, 9, 11, 10, 11, 12, 12, 13, 13, 14, 15, 14, 17, 16, 19, 18, 20, 20, 21, 22, 23, 24, 25, 25, 28, 27, 30, 29, 33, 32, 35, 35, 37, 38, 39

Offset

0,15

Comments

Integer partitions into second or "negative" pentagonal numbers (A005449) .

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

a(n) ~ Gamma(1 + b/d) * zeta(3/2)^(2/3 + b/(3*d)) * d^(1/6 + b/(3*d)) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / (2^(4/3)*d^(1/3))) / (sqrt(3)*2^(7/3 + 2*b/(3*d)) * Pi^(7/6 - b/(6*d)) * n^(7/6 + b/(3*d))) * (1 - (136*d^2 + 120*d*b + 3*b^2*(8 - 3*Pi*zeta(1/2)*zeta(3/2))) / (72 * d^(5/3) * Pi^(1/3) * (2*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/(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(1/prod(k=1, N, (1-x^(k*(3*k+1)/2))))

Crossrefs

Cf. A005449, A095699, A218379, A296238.

Sequence in context: A279222 A276798 A067100 * A074802 A071804 A390843

Adjacent sequences: A296234 A296235 A296236 * A296238 A296239 A296240

Keyword

nonn

Author

Seiichi Manyama, Dec 09 2017

Status

approved