I have used this song for years to teach part of the mathematics of pattern, at every level from 9 year olds to A' level coursework (final year of secondary school).

Someone earlier described the presents on each day as the triangle numbers ( n x n+1)/2.
so on the 12th day my true love gave unto me 12 x 13 /2 i.e. 78 presents.
Discovery and description of this rule is GCSE level work ( Age 16).

The next level of the pattern is the presents so far, ie the running total of triangle numbers and of course this also leads to a further algebraic formula. n x (n+1) x (n+2)/6. so over 12 days 12 x 13 x 14 /6 i.e.364 or one for every day except Christmas Day.

You can show the progressive levels of pattern in a table:
To see the table properly cut and paste it into a word proccessor or spreadsheet.

        1        1        1        1        1        1        1        1        1        1        1        1        n0
Day        1        2        3        4        5        6        7        8        9        10        11        12        n+1
Presents
today        1        3        6        10        15        21        28        36        45        55        66        78        n x (n+1)/2
Presents
so far        1        4        10        20        35        56        84        120        165        220        286        364        n x (n+1) x (n+2)/6
        1        5        15        35        70        126        210        330        495        715        1001        1365        
        1        6        21        56        126        252        462        792        1287        2002        3003        4368        
        1        7        28        84        210        462        924        1716        3003        5005        8008        12376        
        1        8        36        120        330        792        1716        3432        6435        11440        19448        31824        
        1        9        45        165        495        1287        3003        6435        12870        24310        43758        75582        
        1        10        55        220        715        2002        5005        11440        24310        48620        92378        167960        
        1        11        66        286        1001        3003        8008        19448        43758        92378        184756        352716        
        1        12        78        364        1365        4368        12376        31824        75582        167960        352716        705432        

This can be extended infinitely in either direction and is in fact that source of all mathematics, Pascal's Triangle.
You can find a general rule for any element of the table, which involves factorials. You can find the coefficients for binomial expansions (university level maths) but the only rule to generate the whole table is add the number above to the number on the left, starting with 1 surrounded by zeros.


I could go on about this topic for years, in fact I have done so , but I shall shut up now with the observation that the more I study maths and fractals in particular the more I am astounded by the level of complexity yhat can emerge from a simple starting position and a recursive rule.