So if you have Pascal's triangle, I know you can calculate any value in closed form.
1
1 1
1 2 1
1 3 3 1
....
If we let R
be the row number, then we can generate that triangle like this
C(R,0)
C(R,0) C(R,1)
C(R,0) C(R,1) C(R,2)
C(R,0) C(R,1) C(R,2) CR,3)
with the choose function Choose(row#,column#)
but I have a variation on this that looks like this
C(R,0)*C(N,0)
C(R,0)*C(N,0) C(R,1)*C(N,1)
C(R,0)*C(N,0) C(R,1)*C(N,1) C(R,2)*C(N,2)
C(R,0)*C(N,0) C(R,1)*C(N,1) C(R,2)*C(N,2) C(R,2)*C(N,3)
....
So at point in Pascals triangle instead of C(N,Column#)
you have C(R,Column#)*C(N,Column#)
. Where R > Column#
.
So we can calculate any single value in closed form, but if I wanted to calculate a whole row or subset of a row, is there a closed form for that?
r
and an interval integer rangeR
that is a subset of[0,r-1]
and returns the sum of the terms ofR
from the variation of pascals triangle. $\endgroup$