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...

 234 5678 910 1112
1+Σ+Σ+ Σ+Σ+Σ +Σ+Σ+ Σ+Σ+Σ +Σ
d=1d=1d=1 d=1d=1d=1 d=1d=1d=1 d=1d=1
13610 15212836 4555 6678

Click here for my longer blog post about "The 12 Days of Christmas"