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

1,2

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.

Links

Table of n, a(n) for n=1..40.

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

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

Sequence in context: A283143 A174175 A127392 * A258092 A263993 A189585

Adjacent sequences: A235998 A235999 A236000 * A236002 A236003 A236004

Keyword

nonn

Author

Omar E. Pol, Jan 18 2014

Extensions

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

Status

approved