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A078616 a(n) = Sum_{k=0..n} A010815(k). 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

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

PROG

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

CROSSREFS

Cf. A010815, A000326, A246575.

Sequence in context: A175479 A307243 A120530 * A267800 A322980 A267053

Adjacent sequences:  A078613 A078614 A078615 * A078617 A078618 A078619

KEYWORD

sign

AUTHOR

Benoit Cloitre, Dec 10 2002

STATUS

approved

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Last modified August 15 02:47 EDT 2022. Contains 356122 sequences. (Running on oeis4.)