Geodesic
Multiplicative functions generalized binomial coefficients, and generalized Catalan numbers
Journal of integer sequences, Tome 19 (2016) no. 1.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $gcd_{k}$ and $\phi $. We finish by connecting these results to a classical result due to Ward.
Classification : 11B65, 05A10, 11A05
Keywords: multiplicative function, divisible function, generalized binomial coefficient, Kummer's theorem, fuss-Catalan number 
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     author = {Edgar, Tom and Spivey, Michael Z.},
     title = {Multiplicative functions generalized binomial coefficients, and generalized {Catalan} numbers},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
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     number = {1},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2016__19_1_a6/}
}
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Edgar, Tom; Spivey, Michael Z. Multiplicative functions generalized binomial coefficients, and generalized Catalan numbers. Journal of integer sequences, Tome 19 (2016) no. 1. http://geodesic.mathdoc.fr/item/JIS_2016__19_1_a6/