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I encounter a curious sequence $a_i$ which is defined below. I wonder if it has a name and has some closed form.

Let $r, q$ be positive integers. (Assume further that $q$ is a prime power if necessary.) The sequence in question is defined by $$ a_i = \lvert\{\, (e_1, \dotsc, e_r) \in [0, q)^r : e_1 + \dotsb + e_r = i \,\}\rvert \quad (i \geq 0)$$ where $[0, q) = \{0, 1, \dotsc, q - 1\}$.

What I can see is the followings.

  • $a_0 = a_{r(q - 1)} = 1$
  • $a_i = \binom{r}{i}$ if $q = 2$
  • $a_i = 0$ if $i > r(q - 1)$.

(In addition, the identity $\sum_{i = 0}^{r(q - 1)} a_i = q^r$ probably holds judging from the situation I encounter.)

Orat
  • 4,065
  • You can use $(1-x^q)^r(1-x)^{-r}$ as a generating function and expand using Newton-Binomial series. – anon Feb 15 '15 at 20:45
  • You’ll have to do a little translating, but that really is essentially the same question; its $H$ is your $i$, its $RS$ is your $r$, and its $C$ is your $q-1$. This answer, again with different notation, has a much more detailed explanation of where the result comes from. – Brian M. Scott Feb 15 '15 at 20:55
  • @anon I see that its the generating function of my sequence though I haven't get nice closed form yet. – Orat Feb 15 '15 at 21:10
  • @BrianM.Scott Thank you for providing a useful link. I got it. – Orat Feb 15 '15 at 21:42
  • @Taro: You’re welcome. – Brian M. Scott Feb 15 '15 at 21:42

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