Power sums in hyperbolic Pascal triangles

In this paper, we describe a method to determine the power sum of the elements of the rows in the hyperbolic Pascal triangles corresponding to $\{4,q\}$ with $q\ge5$. The method is based on the theory of linear recurrences, and the results are demonstrated by evaluating the $k^{th}$ power sum in the range $2\le k\le 11$.Comment: 13 pages, 1 figure

Point Counts of D_k and Some A_k and E_k Integer Lattices Inside Hypercubes

Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm in the host lattice. The results assume polynomial format as a function of the hypercube edge length.Comment: Merged chapters 9 and 10, and added Table 5 and Conjecture 2

Summations of Linear Recurrent Sequences

We give an extension of Sister Celine's method of proving hypergeometric sum identities that allows it to handle a larger variety of input summands. We then apply this to several problems. Some give new results, and some reprove already known results in an automated way

Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations

For a fixed integer N, and fixed numbers b_1,...,b_N, we consider sequences, the nth term (a_n) of which is the sum of the squares of the terms in the expansion of (b_1 + ... + b_N)^n. In the case all b_i=1, we give a formula for a recurrence relation for the a_n. Otherwise we give an algorithm for finding a recurrence relation. As an application, we give the Picard-Fuchs equations for certain families of elliptic curves.Comment: 20 page

Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers

The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have been obtained and at the same time previously well-known formulas and identities have been generalized. Moreover, on this occasion, we identify new families of polynomials including some families of well-known numbers such as Bernoulli numbers, Euler numbers, Stirling numbers, Franel numbers, Catalan numbers, Changhee numbers, Daehee numbers and the others, and also for the polynomials such as the Legendre polynomials, Michael Vowe polynomial, the Mirimanoff polynomial, Golombek type polynomials, and the others. We also give both Riemann and $p$-adic integral representations of these polynomials. Finally, we give combinatorial interpretations of these new families of numbers, polynomials and finite sums of the powers of binomial coefficients. We also give open questions for ordinary generating functions for these numbers.Comment: 36 page

Calculations relating to some special Harmonic numbers

We report on the results of a computer search for primes $p$ which divide an Harmonic number $H_{\lfloor p/N \rfloor}$ with small $N > 1$

Identities in character tables of SnS_n

In the classic "Concrete Math", by Graham, Patashnik and Knuth, it is stated that "The numbers in Pascal's triangle satisfy, practically speaking, infinitely many identities, so it is not too surprising that we can find some surprising relationships by looking closely." The aim of this note is to indicate that a similar statement seems to hold for the character tables of the symmetric groups $S_n$. Just as important, it is a case-study in using a computer algebra system to prove deep identities, way beyond the ability of mere humans. This article is accomanied by a Maple pacgage, Sn, and ample output, avaialble from the webpage http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/sn.html .Comment: 9 page

On a family of recurrences that includes the Fibonacci and the Narayana recurrences

We survey and prove properties a family of recurrences bears in relation to integer representations, compositions, the Pascal triangle, sums of digits, Nim games and Beatty sequences.Comment: 18 page

The Filbert Matrix

A Filbert matrix is a matrix whose (i,j) entry is 1/F_(i+j-1), where F_n is the nth Fibonacci number. The inverse of the n by n Filbert matrix resembles the inverse of the n by n Hilbert matrix, and we prove that it shares the property of having integer entries. We prove that the matrix formed by replacing the Fibonacci numbers with the Fibonacci polynomials has entries which are integer polynomials. We also prove that certain Hankel matrices of reciprocals of binomial coefficients have integer entries, and we conjecture that the corresponding matrices based on Fibonomial coefficients have integer entries. Our method is to give explicit formulae for the inverses.Comment: 9 pages; submitted to The Fibonacci Quarterl

On the binomial interpolated triangles

The binomial interpolated transform of a sequence is a generalization of the well-known binomial transform. We examine a Pascal-like triangle, on which a binomial interpolated transform works between the left and right diagonals, focusing on binary recurrences. We give the sums of the elements in rows and in rising diagonals, further we define two special classes of these arithmetical triangles.Comment: 16 pages, 8 figure