Congruences involving \(g_n(x)=\sum\limits_{k=0}^n\dbinom{n}{k}^2\dbinom{2k}{k}x^k\)

DOI10.1007/s11139-015-9727-3zbMath1366.11006OpenAlexW2142504176MaRDI QIDQ312434

Zhi-Wei Sun

Publication date: 15 September 2016

Published in: The Ramanujan Journal (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s11139-015-9727-3

zbMATH Keywords

congruencesbinomial coefficientsFranel numbersApéry numbers

Mathematics Subject Classification ID

Factorials, binomial coefficients, combinatorial functions (05A10) Binomial coefficients; factorials; (q)-identities (11B65) Congruences; primitive roots; residue systems (11A07)

Related Items (11)

On congruences involving Apéry numbers ⋮ Telescoping method and congruences for double sums ⋮ Proof of some congruences conjectured by Z.-W. Sun ⋮ Congruences for Apéry numbers βn =∑k=0nn k2n+k k ⋮ New series for powers of \(\pi\) and related congruences ⋮ On two congruences involving Apéry and Franel numbers ⋮ Divisibility results on Franel numbers and related polynomials ⋮ Supercongruences involving Lucas sequences ⋮ Hankel-type determinants for some combinatorial sequences ⋮ Proof of a conjecture involving Sun polynomials ⋮ Ramanujan-type supercongruences involving Almkvist-Zudilin numbers

Uses Software

• OEIS

Cites Work

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• Congruences for Franel numbers
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• On sums of Apéry polynomials and related congruences
• On congruences related to central binomial coefficients
• More congruences for central binomial coefficients
• Connections between \(p=x^2+3y^2\) and Franel numbers
• New congruences for central binomial coefficients
• A proof that Euler missed. Apéry's proof of the irrationality of \(\zeta(3)\). An informal report
• Binomial identities -- combinatorial and algorithmic aspects
• Recurrences and Legendre transform
• Super congruences and Euler numbers
• \(p\)-adic congruences motivated by series
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• On sums of binomial coefficients modulo p²
• ON SOME NEW CONGRUENCES FOR BINOMIAL COEFFICIENTS
• Arithmetic theory of harmonic numbers
• WOLSTENHOLME TYPE THEOREM FOR MULTIPLE HARMONIC SUMS
• Supercongruences for the Catalan-Larcombe-French numbers

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