A292518 Expansion of Product_{k>=1} (1 - x^(k*(k+1)/2)).

1, -1, 0, -1, 1, 0, -1, 1, 0, 1, -2, 1, 0, 1, -1, -1, 2, -1, 1, -2, 1, 0, 0, 0, 0, 1, -1, 1, -3, 2, -1, 2, -1, 0, 1, -1, 0, -2, 3, -1, 1, -2, 1, 1, -2, 0, 0, 2, 0, -1, 0, 2, -2, -1, -1, 1, 2, -1, 1, -1, 1, -2, 1, -2, 3, 1, -2, 0, -2, 3, -1, -1, 0, 3, -1, 0, -2, 1, 0, -3, 2, 2, 1, -1, -1, 0, 0, -1, 0, 2, -1

Offset: 0

Author

Ilya Gutkovskiy, Sep 18 2017

Keywords

• sign

Comments

Convolution inverse of A007294.

The difference between the number of partitions of n into an even number of distinct triangular numbers and the number of partitions of n into an odd number of distinct triangular numbers.

Euler transform of {-1 if n is a triangular number else 0, n > 0} = -A010054. - Gus Wiseman, Oct 22 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Index entries for related partition-counting sequences

Crossrefs

Product_{k>=1} (1 - x^(k*((m-2)*k-(m-4))/2)): this sequence (m=3), A276516 (m=4), A305355 (m=5).

Cf. A007294, A010054, A024940, A280366, A320767, A320784.

Programs

Mathematica

nmax = 90; CoefficientList[Series[Product[1 - x^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 - x^(k*(k+1)/2)).