WeightedSize on IntegerPartition

The sum of the parts weighted by their row, counting rows from 0.

On IntegerPartition

Called as CombinatorialStat(x, "WeightedSize"). The head's page has its examples.

Live

FindStat St000185

The weighted size of a partition.

Object	Value
`[]`	`0`
`[1]`	`0`
`[2]`	`0`
`[3]`	`0`
`[4]`	`0`
`[5]`	`0`
`[6]`	`0`
`[7]`	`0`

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000185 has the rest.