On the first day there was one gift - one partridge in a pear tree.But the total gifts on the second day would be the sum of the series between two (turtles doves) and one (a partridge in a pear tree). Three gifts total (indicated in the bottom row).

Gifts on the third day start at three (french hens), but again, the total is the sum of the numbers between three and one (partridge in a pear tree).

And so on...

2 3 4 5 6 7 8 9 10 11 12 1 + Σ + Σ + Σ + Σ + Σ + Σ + Σ + Σ + Σ + Σ + Σ d=1 d=1 d=1 d=1 d=1 d=1 d=1 d=1 d=1 d=1 d=1 1 3 6 10 15 21 28 36 45 55 66 78