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For k = 1,2,... let $H_{k}$ denote the harmonic number $\sum_{j = 1}^{k} 1 / j$ . In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have $\sum_{k = 1}^{p - 1} (H_{k}) / (k 2^{k}) \equiv 7 / 24 p B_{p - 3} (m o d p ²)$ , $\sum_{k = 1}^{p - 1} (H_{k, 2}) / (k 2^{k}) \equiv - 3 / 8 B_{p - 3} (m o d p)$ , and $\sum_{k = 1}^{p - 1} (H ²_{k, 2 n}) / (k^{2 n}) \equiv ((\binom{6 n + 1}{2 n - 1}) + n) / (6 n + 1) p B_{p - 1 - 6 n} (m o d p ²)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k, m} : = \sum_{j = 1}^{k} 1 / (j^{m})$ .

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Displaying 61 – 80 of 353

An algebraic summation over the set of partitions and some strange evaluations

An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral indentities.

An area-to-inv bijection between Dyck paths and 312-avoiding permutations.

An extension of the Foata map to standard Young tableaux.

An extension of the Ramanujan-Bailey formula. (Une extension d'une formule de Ramanujan-Bailey.)

An identity for fixed points of permutations.

An identity generator: basic commutators.

An infinite family of dual sequence identities.

An invariant sum related to record statistics.

An inversion formula, matrix functions, combinatorial identities and graphs

Arithmetic theory of harmonic numbers (II)

Around The Nuber Of Chains In Partitive Sets

Around the Razumov-Stroganov conjecture: proof of a multi-parameter sum rule.

Ascent sequences and upper triangular matrices containing non-negative integers.

Asymptotic analysis of Ramanujan pairs.

Asymptotic analysis of Ramanujan pairs. (Short Communication).

Aztec diamonds and Baxter permutations.

Balls on the lawn.

Baxter permutations and plane bipolar orientations.

Bender's generalized q-binominal Vandermonde convolution.

Currently displaying 61 – 80 of 353