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Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that
$T_{p-1} ≡ (p/3) 3^{p-1} (mod p²)$,
where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1 + x + x^{-1})ⁿ$. We also prove three congruences modulo p³ conjectured by Sun, one of which is
$∑_{k=0}^{p-1} \binom{p-1}{k}\binom{2k}{k} ((-1)^k - (-3)^{-k}) ≡ (p/3)(3^{p-1} - 1) (mod p³)$.
In addition, we get some new combinatorial identities.
$T_{p-1} ≡ (p/3) 3^{p-1} (mod p²)$,
where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1 + x + x^{-1})ⁿ$. We also prove three congruences modulo p³ conjectured by Sun, one of which is
$∑_{k=0}^{p-1} \binom{p-1}{k}\binom{2k}{k} ((-1)^k - (-3)^{-k}) ≡ (p/3)(3^{p-1} - 1) (mod p³)$.
In addition, we get some new combinatorial identities.
Słowa kluczowe
Kategorie tematyczne
- 05A19: Combinatorial identities, bijective combinatorics
- 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
- 11B65: Binomial coefficients; factorials; q -identities
- 05A10: Factorials, binomial coefficients, combinatorial functions
- 11A07: Congruences; primitive roots; residue systems
Czasopismo
Rocznik
Tom
Numer
Strony
127-136
Opis fizyczny
Daty
wydano
2015
Twórcy
autor
- Department of Applied Mathematics, Nanjing Audit University, Nanjing 211815, People's Republic of China
autor
- Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm139-1-8