A078616 a(n) = Sum_{k=0..n} A010815(k).

1, 0, -1, -1, -1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Author

Benoit Cloitre, Dec 10 2002

Keywords

• sign

Comments

To construct the sequence: a(0)=1, a(1)=0, then (2*1+1) (-1)'s followed by 2 0's, followed by (2*2+1) 1's, followed by 3 0's, followed by (2*3+1) (-1)'s, etc.

From George Beck, May 05 2017: (Start)

a(n) = (Number of ones in the distinct partitions of n with an odd number of parts) - (number of ones in the distinct partitions of n with an even number of parts) (conjectured).

The partial sums give A246575. (End) [corrected by Ilya Gutkovskiy, Aug 18 2018]

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Mircea Merca, Higher-order differences and higher-order partial sums of Euler's partition function, 2018.

Crossrefs

Cf. A010815, A000326, A246575.

Programs

PARI

a(n)=polcoeff(eta(x)/(1-x)+O(x^n),n)

Formula

For m > 0, a(k)=0 if A000326(m) <= k < A000326(m) + m; a(k)=(-1)^m if A000326(m) + m <= k < A000326(m+1).

G.f.: eta(x)/(1-x). - Benoit Cloitre, Jan 31 2004

G.f.: exp(-Sum_{k>=1} (sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 18 2018