The numbers in the first column are: $1, 1, -3, 15, -105, 945, ...$
Except at the beginning, these alternate in sign and are found by multiplying
all the odd numbers up to a certain point.
$1, 1, -1*3, 1*3*5, -1*3*5*7, 1*3*5*7*9, ...$
Like @Gareth-McCaughan points out, these have a notation using the double
factorial symbol. $(-1)!!=-1$, so the first number is $-(-1)!!$.
The second number is $1!!$, the third is $-3!!$, etc.
There are some other patterns that can be
spotted on the diagonals such as all the 1's on the first diagonal.
The second and third diagonal
also seem to follow a simple pattern.
The key to finding all the other numbers
is the diagonals.
A pattern can be seen if you take repeated differences.
The first diagonal is: $1, 1, 1, 1, 1, 1, ...$
the differences between terms in this sequence are: $0, 0, 0, 0, ...$
The second diagonal is: $1, 2, 3, 4, 5, ...$
and the differences between successive terms are: $1, 1, 1, 1, 1, ...$
the differences of the differences (i.e. the second differences) are:
$0, 0, 0, 0, 0, ...$
The third diagonal is: $-3, -7, -12, -18, -25, -33, ...$
the differences are: $-4, -5, -6, -7, -8, ...$
the second differences are: $-1, -1, -1, -1, -1, ...$
the third differences are: $0, 0, 0, 0, 0, ...$
The fourth diagonal is: $15, 40, 78, 132, 205, ...$
differences: $25, 38, 54, 72, ...$
That doesn't look so promising, but maybe if we take the differences again?
second differences: $13, 16, 19, 22, ...$
That looks better; the successive terms differ by 3.
third differences: $3, 3, 3, 3, 3, ...$
fourth differences: $0, 0, 0, 0, 0,...$
The numbers are getting bigger, but there seems to be something to this.
Let's keep pressing on.
The fifth diagonal is: $-105, -315, -693, -1317, -2280, -3690,...$
differences: $-210, -378, -624, -963, -1410,...$
second differences: $-168, -246, -339, -447, ...$
third differences: $-78, -93, -108, ...$
fourth differences: $-15, -15, -15, -15, ...$
fifth differences: $0, 0, 0, 0, 0, ...$
Maybe in the sixth diagonal, the sixth differences will all be 0? Let's see.
$945, 3150, 7749, 16416, 31470, 55980, 93870,...$
differences: $2205, 4599, 8667, 15054, 24510, 37890, ...$
second differences: $2394, 4068, 6387, 9456, 13380, ...$
third differences: $1674, 2319, 3069, 3924, ...$
fourth differences: $645, 750, 855, ...$
fifth differences: $105, 105, 105, ....$
sixth differences: $0, 0,0,0,0,...$
So, that's the pattern.
In the $k^{th}$ diagonal, the $k^{th}$ successive differences are 0. The
$(k-1)^{th}$ differences in the cases calculated above
(for diagonals 3 through 6) were: $-1, 3, -15, 105$.
Look familiar? These are the products of the odd
numbers again just like in the first column!
They have to be those numbers to
make the pattern of the differences and everything else work out.
Knowing everything about the differences now, you can work backwards
to find out what all the original numbers in the triangle have to be.
All those calculations made me hungry. Luckily, my wife is making the world's best spaghetti using a recipe she and her friend stole from some rich guy in a local mansion. It's a funny story how they got the recipe.