Series Representation of Power Function
Résumé
In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence $\href{https://oeis.org/A287326}{A287326}$ in OEIS. The main property of $\href{https://oeis.org/A287326}{A287326}$ that it returns a perfect cube $n$ as sum of $n$-th row terms over $k$, such that $0\leq k\leq n-1$ or $1\leq k\leq n$, by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers $m=5,7$ as row items sum and generalized obtained results in order to receive every odd-powered monomial $n^{2m+1}, \ m\geq0$ as sum of row terms of corresponding triangle. In other words, in this manuscript are found and discussed the polynomials $D_m(n,k)$ and $U_m(n,k)$, such that, when being summed up over $k$ in some range with respect to $m$ and $n$ returns the monomial $n^{2m+1}$.
Mots clés
Series Representation
Power function
monomial
Binomial Theorem
Multinomial theorem
Worpitzky Identity
Stirling numbers of second kind
Faulhaber’s sum
finite difference
Faulhaber’s formula
central factorial numbers
binomial coefficients
Binomial distribution
binomial transform
Bernoulli numbers
OEIS
multinomial coefficients
Origine : Fichiers produits par l'(les) auteur(s)
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