Finding an expression to represent this pattern

Question

Is there a method to find a math expression for a given pattern? I have this pattern and I am very curious to find out how can I generate it.

0 1 -1x 3 -4x 3x^2 5 -9x 9x^2 -5x^3 7 -16x 21.6x^2 -16x^3 7x^4 9 -25x 42.86x^2 -42.86x^3 25x^4 -9x^5 11 -36x 75x^2 -95.24x^3 75x^4 -36x^5 11x^6

Answer 1

• The n-th row has n elements, starting with the first.
• The signs alternate on each row, and each row starts with a $+$.
• The first and last element of each row (starting with the second) host the odd numbers $2n-3$.
• The second and next-to-last element on each row (starting with the third) host the perfect squares $(n-1)^2$. Notice that the differences of each two consecutive squares are exactly the odd numbers mentioned before.
• The third and the next-to-next-to-last element on each row (starting with the fifth) host a number whose absolute value is very close to the arithmetic mean of the absolute values of the one to the side, and the one right above it, minus $4$.
• As far as the element in the middle of the seventh row is concerned, I need to see more rows before determining a specific pattern.
• Hope this helps !