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A236001
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Sum of positive ranks of all overpartitions of n.
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7
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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
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The sum of positive ranks of all overpartitions of 4 is 3+3+1+1+1+1 = 10 so a(4) = 10.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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