So I can build a triangle by detailing the coefficients of the two above series:
I can keep writing, notice that the subscript for $a$ doesn't change. It is easy to write these coefficients in a triangle which I call the Cauchy triangle.
Recently, however, I encounter a problem that asks me to deal with the product of four power series instead of two. So I wish to build up a triangle that is similar to the case of two power series.
I write the four power series as followed:
$A\cdot B\cdot C\cdot D=a_0b_0c_0d_0+(a_1b_0c_0d_0+a_0b_1c_0d_0+a_0b_0c_1d_0)x+(a_0b_2c_0d_0+a_2b_0c_0d_0+a_0b_0c_2d_0+a_0b_0c_0d_2 )x^2...$
This is the best I can try so far, is there a method to exhaustively list all these elements into a triangle like the case of two power series? I want a complete triangle to fully count the coefficients.
Please don't respond to this thread with full sigma notation, as I haven't mastered the procedure of manipulating these symbols yet.