Integrating across Pascal's triangle

Sums across the rows of Pascal's triangle yield n2 while certain diagonal sums yield the Fibonacci numbers which are asymptotic to ϕn where ϕ is the golden ratio. Sums across other diagonals yield quantities asymptotic to cn where c depends on the directions of the diagonals. We generalize this to the continuous case. Using the gamma function, we generalize the binomial coefficients to real variables and thus form a generalization of Pascal's triangle. Integration over various families of lines and curves yields quantities asymptotic to cx where c is determined by the family and x is a parameter. Finally, we revisit the discrete case to get results on sums along curves.