A236001 Sum of positive ranks of all overpartitions of n.

0, 2, 4, 10, 20, 36, 64, 110, 180, 288, 452, 696, 1052, 1568, 2304, 3346, 4808, 6838, 9636, 13464, 18664, 25684, 35104, 47672, 64348, 86368, 115304, 153152, 202452, 266404, 349032, 455406, 591856, 766284, 988544, 1270862, 1628380, 2079828, 2648296, 3362180

Offset: 1

Author

Omar E. Pol, Jan 18 2014

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

Cf. A015128, A209616, A235790, A235792, A235797, A235798, A236000.

Programs

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

Extensions

Terms a(7) and beyond from Andrew Howroyd, Feb 19 2020