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Subjects
11-XX Number theory
11Bxx Sequences and sets
11B65 Binomial coefficients; factorials; $q$ -identities

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Displaying 1 – 17 of 17

Tabulka binomických koefficientů ${(\frac{1}{n})}_{r}$

Čeněk Jarolímek (1889)

Časopis pro pěstování mathematiky a fysiky

The Akiyama-Tanigawa algorithm for Carlitz's $q$ -Bernoulli numbers.

Zeng, Jiang (2006)

Integers

The Euler series transformation and the binomial identities of Ljunggren, Munarini and Simons.

Boyadzhiev, Khristo N. (2010)

Integers

The function $_{v} M_{m} (s; a, z)$ and some well-known sequences.

Petojević, Aleksandar (2002)

Journal of Integer Sequences [electronic only]

The $k$ -binomial transforms and the Hankel transform.

Spivey, Michael Z., Steil, Laura L. (2006)

Journal of Integer Sequences [electronic only]

The Kazandzidis supercongruences. A simple proof and an application

Alain Robert, Maxime Zuber (1995)

Rendiconti del Seminario Matematico della Università di Padova

The Lagrange inversion formula and divisibility properties.

Woan, Wen-jin (2007)

Journal of Integer Sequences [electronic only]

The last digit of $(\binom{2 n}{n})$ and $\sum (\binom{n}{i}) (\binom{2 n - 2 i}{n - 1})$ .

Shur, Walter (1997)

The Electronic Journal of Combinatorics [electronic only]

The number of binomial coefficients in residue classes modulo p and $p^{2}$ .

William A. Webb (1990)

Colloquium Mathematicae

The number of topologies on a finite set.

Benoumhani, Moussa (2006)

Journal of Integer Sequences [electronic only]

The p-adic valuation of k-central binomial coefficients

Armin Straub, Victor H. Moll, Tewodros Amdeberhan (2009)

Acta Arithmetica

Transforming recurrent sequences by using the binomial and invert operators.

Barbero, Stefano, Cerruti, Umberto, Murru, Nadir (2010)

Journal of Integer Sequences [electronic only]

Triangular numbers in geometric progression.

Chen, Yong-Gao, Fang, Jin-Hui (2007)

Integers

Truncating binomial series with symbolic summation.

Paule, Peter, Schneider, Carsten (2007)

Integers

Truncations of Gauss' square exponent theorem

Ji-Cai Liu, Shan-Shan Zhao (2022)

Czechoslovak Mathematical Journal

We establish two truncations of Gauss’ square exponent theorem and a finite extension of Euler’s identity. For instance, we prove that for any positive integer $n$ , $\sum_{k = 0}^{n} {(- 1)}^{k} [\begin{matrix} 2 n - k \\ k \end{matrix}] {(q; q^{2})}_{n - k} q^{(\binom{k + 1}{2})} = \sum_{k = - n}^{n} {(- 1)}^{k} q^{k^{2}},$ where $[\begin{matrix} n \\ m \end{matrix}] = \prod_{k = 1}^{m} \frac{1 - q^{n - k + 1}}{1 - q^{k}} and {(a; q)}_{n} = \prod_{k = 0}^{n - 1} (1 - a q^{k}) .$

Two $q$ -identities from the theory of fountains and histograms proved with a tri-diagonal determinant.

Prodinger, Helmut (2005)

Integers

Two very short proofs of combinatorial identity.

Anglani, Roberto, Barile, Margherita (2005)

Integers

Currently displaying 1 – 17 of 17

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