OFFSET
1,2
COMMENTS
Consider here that the rank of a overpartition is the largest part minus the number of parts (the same idea as the Dyson's rank of a partition).
It appears that the sum of all ranks of all overpartitions of n is equal to zero.
The equivalent sequence for partitions is A209616.
EXAMPLE
For n = 4 we have:
---------------------------
Overpartitions
of 4 Rank
---------------------------
4 4 - 1 = 3
4 4 - 1 = 3
2+2 2 - 2 = 0
2+2 2 - 2 = 0
3+1 3 - 2 = 1
3+1 3 - 2 = 1
3+1 3 - 2 = 1
3+1 3 - 2 = 1
2+1+1 2 - 3 = -1
2+1+1 2 - 3 = -1
2+1+1 2 - 3 = -1
2+1+1 2 - 3 = -1
1+1+1+1 1 - 4 = -3
1+1+1+1 1 - 4 = -3
---------------------------
The sum of positive ranks of all overpartitions of 4 is 3+3+1+1+1+1 = 10 so a(4) = 10.
PROG
(PARI) a(n)={my(s=0); forpart(p=n, my(r=p[#p]-#p); if(r>0, s+=r*2^#Set(p))); s} \\ Andrew Howroyd, Feb 19 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jan 18 2014
EXTENSIONS
Terms a(7) and beyond from Andrew Howroyd, Feb 19 2020
STATUS
approved