Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.1

Congruences of Multiple Sums Involving Sequences Invariant Under the Binomial Transform


Sandro Mattarei
Dipartimento di Matematica
Università di Trento
via Sommarive, 14
38100 Trento
Italy

Roberto Tauraso
Dipartimento di Matematica
Università di Roma "Tor Vergata"
via della Ricerca Scientifica
00133 Roma
Italy

Abstract:

We prove several congruences modulo a power of a prime, such as

\begin{displaymath} \sum_{0<k_1<\cdots<k_{n}<p}\genfrac{(}{)}{}{}{p-k_{n}}{3} \f... ...eft({1/3}\right) \pmod{p}, &\text{if $n$\ is even,} \end{cases}\end{displaymath}

where $ n$ is a positive integer and $ p$ is a prime such that $ p>\max(n+1,3)$.

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Received November 5 2009; revised version received April 16 2010. Published in Journal of Integer Sequences, April 16 2010.

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