A393987 Expansion of g.f.: Product_{k>=1} (1 + x^(k*(k+3)/2)).

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 2, 1, 2, 1, 1, 1, 0, 3, 1, 3, 1, 1, 1, 1, 3, 1, 3, 2, 1, 3, 1, 2, 4, 2, 3, 2, 1, 3, 3, 3, 2, 4, 1, 4, 3, 2, 5, 3, 4, 3, 3, 3, 5, 4, 5, 3, 4, 4, 3, 6, 3, 6, 6, 4, 6, 4, 5, 5, 7, 4, 6

Offset

0,15

Comments

a(n) is the number of partitions of n into distinct parts of the form k*(k+3)/2.

Links

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

Formula

a(n) ~ exp(3*Pi^(1/3) * ((sqrt(2)-1)*Zeta(3/2))^(2/3) * n^(1/3) / 2^(4/3)) * ((sqrt(2)-1)*Zeta(3/2))^(1/3) / (2^(8/3) * sqrt(3) * Pi^(1/3) * n^(5/6)).

Mathematica

nmax = 200; CoefficientList[Series[Product[1 + x^(k*(k+3)/2), {k, 1, Floor[Sqrt[9+8*nmax]/2]}], {x, 0, nmax}], x]

Crossrefs

Cf. A024940, A319799, A393989.

Sequence in context: A254886 A030219 A287345 * A260675 A035147 A101673

Adjacent sequences: A393984 A393985 A393986 * A393988 A393989 A393990

Keyword

nonn

Author

Vaclav Kotesovec, Mar 06 2026

Status

approved