Journal of Integer Sequences, Vol. 16 (2013), Article 13.1.3

Restricted Weighted Integer Compositions and Extended Binomial Coefficients

Steffen Eger
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213


We prove a simple relationship between extended binomial coefficients natural extensions of the well-known binomial coefficients and weighted restricted integer compositions. Moreover, we give a very useful interpretation of extended binomial coefficients as representing distributions of sums of independent discrete random variables. We apply our results, e.g., to determine the distribution of the sum of k logarithmically distributed random variables, and to determine the distribution, specifying all moments, of the random variable whose values are part-products of random restricted integer compositions. Based on our findings and using the central limit theorem, we also give generalized Stirling formulae for central extended binomial coefficients. We enlarge the list of known properties of extended binomial coefficients.

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(Concerned with sequence A002426.)

Received April 16 2012; revised versions received April 23 2012; September 7 2012; October 13 2012; January 3 2013. Published in Journal of Integer Sequences, January 4 2013.

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