Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6

Multiplicative Functions Generalized Binomial Coefficients, and Generalized Catalan Numbers

Tom Edgar
Department of Mathematics
Pacific Lutheran University
Tacoma, WA 98447-0003

Michael Z. Spivey
Department of Mathematics and Computer Science
University of Puget Sound
Tacoma, WA 98416-1043


We investigate generalized binomial coefficients of multiplicative functions. We provide a formula for these coefficients and use this formula to prove that the coefficients are always integral if the function is also a divisible function. Furthermore, we prove that multiplicative and divisible functions have integral generalized Fuss-Catalan numbers. Along the way, we include some results about specific multiplicative functions such as gcdk and φ. We finish by connecting these results to a classical result due to Ward.

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(Concerned with sequences A000010 A000027 A000034 A000045 A000129 A000188 A000265 A000290 A000578 A001615 A002445 A002618 A003557 A003958 A003959 A003961 A006519 A007318 A007434 A007947 A008459 A008833 A010048 A022998 A026741 A034444 A038500 A038502 A046951 A047994 A048804 A059376 A059377 A059378 A060594 A060904 A061142 A064549 A069091 A069092 A069093 A069094 A069095 A082907 A089128 A099927 A108951 A109007 A109008 A109009 A109010 A109011 A132739 A141056 A173557 A181543 A191414 A214281 A214681 A214682 A214685 A234957 A234959 A238453 A238498 A238688 A238743 A238754 A239633 A239702 A242603 A242849 A243756 A246465 A254609 A254730 A255914 A255915

Received September 17 2015; revised versions received December 10 2015; December 17 2015. Published in Journal of Integer Sequences, December 18 2015.

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