Formulas for the number of binomial coefficients divisible by a fixed power of a prime
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- by F. T. Howard PDF
- Proc. Amer. Math. Soc. 37 (1973), 358-362 Request permission
Abstract:
Define ${\theta _j}(n)$ as the number of binomial coefficients $\binom {n}{s}$ divisible by exactly ${p^j}$. A formula for ${\theta _2}(n)$ is found, for all $n$, and formulas for ${\theta _j}(n)$ for $n = a{p^k} + b{p^r}$ and $n = {c_1}{p^{{k_1}}} + \cdots + {c_m}{p^{{k_m}}}$ (${k_1} \geqq j$, ${k_{i + 1}} - {k_i} \geqq j$ for $i = 1$, …, $m - 1$) are derived.References
- L. Carlitz, The number of binomial coefficients divisible by a fixed power of a prime, Rend. Circ. Mat. Palermo (2) 16 (1967), 299–320. MR 249308, DOI 10.1007/BF02843799 L. E. Dickson, History of the theory of numbers. Vol. 1, Publication no. 256, Carnegie Institution of Washington, Washington, D.C., 1919.
- N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 (1947), 589–592. MR 23257, DOI 10.2307/2304500
- F. T. Howard, The number of binomial coefficients divisible by a fixed power of $2$, Proc. Amer. Math. Soc. 29 (1971), 236–242. MR 302459, DOI 10.1090/S0002-9939-1971-0302459-9
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 37 (1973), 358-362
- MSC: Primary 05A10; Secondary 10A99
- DOI: https://doi.org/10.1090/S0002-9939-1973-0309737-X
- MathSciNet review: 0309737