Yes, "week" should be "year". I was quoting from memory.

If I had to make a guess, it would be that this song dates to the period 1910-1930, when grisly humor involving children was popular; cf. the "little Willy" rhymes (according to George Orwell, one Harry Graham started them off with a book called _Ruthless Rhymes for Heartless Homes_) and some of the the "little Audrey" jokes. Here are a few examples (the first quoted by Orwell):

Poor little Willy is crying so sore,
A sad little boy is he,
For he's broken his little sister's neck
And he'll have no jam for tea.

Willy, in one of his bright new sashes,
Fell in the fire and was burned to ashes.
Now, altho the room grows chilly,
I haven't the heart to poke poor Willy.

Little Willy from the mirror
Licked the mercury right off,
Thinking, in his childish error,
It would cure the whooping cough.
At the funeral, his mother
Smartly said to Mrs Brown,
"'Twas a chilly day for Willy
When the mercury went down."

Mother heard her children scream,
So she threw them in the stream,
Saying, as she drowned the third,
"Children should be seen, not heard."

Little Audrey was playing with matches and burned the house down. Her mother said, "Just wait till your father comes home -- you'll really catch it then". But little Audrey just laughed and laughed, because she knew her father had come home early and had been sleeping upstairs.

The period mentioned perhaps roughly corresponds to the time when, in the industrial countries, public health (immunization & sanitation) had become good enough that deaths of children were uncommon. Up to that time, most families had lost children, and had to keep a stiff upper lip about it. The flagrant tastelessness of the jokes may reflect inner conflict due to the softening of attitudes resulting from the new opportunities to care more about children.

On the other hand, children have always enjoyed being gross, one way or another.