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# 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.

Central Difference, Gauss's Forward Formula

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## 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.

## Referenced on Wolfram|Alpha

Gauss's Backward Formula

## Cite this as:

Weisstein, Eric W. "Gauss's Backward Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GausssBackwardFormula.html