Gauss's Backward Formula

This is sometimes knows as the "bars and stars" method. Suppose a recipe called for 5 pinches of spice, out of 9 spices. Each possibility is an arrangement of 5 spices (stars) and 9 dividers between categories (bars). The number of possibilities is .  means you use spices 1, 1, 5, 6, and 9.

(1)

for , where  is the central difference and

			(2)
			(3)

where  is a binomial coefficient.

REFERENCES:

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987.

Whittaker, E. T. and Robinson, G. "The Newton-Gauss Backward Formula." §22 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 37-38, 1967.