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Let $p$ be a prime and $a_0,\ldots,a_n\geq 0$ be integers. Define $$ S(a_0,\ldots,a_n)=\sum_{k=0}^n a_k\binom{n}{k}. $$

I am trying to find out how much we know about $$ v_p(S(a_0,\ldots,a_n)), $$ where $v_p$ is the $p-$adic valuation, or identities involving $S(a_0,\ldots,a_n)$.

Here are a few classical ones, $S(1,x,\ldots,x^n)=(1+x)^n$ and $S(0,1,\ldots,n)=n2^{n-1}$.

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  • $\begingroup$ $v_p$ is $p$-adic valuation? $\endgroup$
    – Sam Hopkins
    Feb 27, 2020 at 16:52
  • $\begingroup$ Yes. I've added that to the original post. Thx! $\endgroup$
    – rpc
    Feb 27, 2020 at 18:54
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    $\begingroup$ It is known is that if $f(x)$ is the exponential generating function of the sequence $a_n$, then $f(x)\cdot e^x$ is the exponential generating function of $S(a_0,...,a_n)$. This observation brings a lot of classical cases. $\endgroup$
    – René Gy
    Feb 27, 2020 at 19:55
  • $\begingroup$ An example with $n=pd$, $\nu_p(S) = 2d-1$, and $a_k$ depending only on $d$ but not $p$ is given in my paper: arxiv.org/abs/1602.02632 $\endgroup$
    – Max Alekseyev
    Feb 28, 2020 at 1:59
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    $\begingroup$ Needs details or clarity. Arbitrary number can be represented in this form $a_0=N$, $a_1=...=a_n=0.$ $\endgroup$
    – Alexey Ustinov
    Feb 28, 2020 at 4:13

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