On the expansion of the power of any polynomial 1+x+x^2+x^3+x^4+etc

Sum of powers 1^p+...+n^p, with n and p being natural numbers and n>=1, can be expressed as a polynomial function of n of degree p+1. Such representations are often called Faulhaber formulae. A simple recursive algorithm for computing coefficients of Faulhaber formulae is presented. The correctness of the algorithm is proved by giving a recurrence relation on Faulhaber formulae.

We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is an introduction. In the second section we derive an explicit formula for F. From the expression for the power function we obtain a number theory result. Then we derive a formula which shows that the case of arbitrary m may be reduced...

In this article a new method of generating sums of like powers is presented.

In this paper one extends the binomial and trinomial coefficients to the concept of 'k-nomial' coefficients, and one obtains some properties of these. As an application one generalizes Pascal's triangle.

In this paper, we provide a general framework for obtaining the formula for nested summation of powers of natural numbers. We define a special triangular array of numbers from which we can obtain the formula for nested summation of natural numbers at any particular power. Binomial coefficients play a key role in nesting. Our framework is very simple to understand.

Here Euler notes the recursive relation for the general binomial coefficients, by assuming that (1+x)^a can be expanded in a power series.

We geometrically prove that in a d-dimensional cube with edges of length n, the number of particular d-dimensional tetrahedrons are given by Eulerian numbers. These tetrahedrons tassellate the cube, In this way the sum of the cubes are the sums of the tetrahedrons, whose calculation is trivial.

Early 17th-century mathematical publications of Johann Faulhaber contain some remarkable theorems, such as the fact that the $r$-fold summation of $1^m,2^m,...,n^m$ is a polynomial in $n(n+r)$ when $m$ is a positive odd number. The present paper explores a computation-based approach by which Faulhaber may well have discovered such results, and solves a 360-year-old riddle that Faulhaber presented to his readers. It also shows that similar results hold when we express the sums...

We consider formal power series $ f(x) = a_1 x + a_2 x^2 + \cdots $ $(a_1 \neq 0)$, with coefficients in a field. We revisit the classical subject of iteration of formal power series, the n-fold composition $f^{(n)}(x)=f(f(\cdots (f(x)\cdots))=f^{(n-1)}(f(x))=\sum\limits_{k=1}^{\infty}f_k^{(n)}x^k$ for $n=2,3,\ldots$, where $f^{(1)}(x)=f(x)$. The study of this was begun, and the coefficients $f_k^{(n)}$ where calculated assuming $a_1=1$, by Schr\"oder in 1871. The major resul...

This paper is about the product z^q/(1 - z)^(q + 1)(1 + (z/(1 - z)))^p, Euler gives the Talylor-Series and takes a closer look at the coefficient.