Chabert, Jean-Luc; Cahen, Paul-Jean Old problems and new questions around integer-valued polynomials and factorial sequences. (English) Zbl 1117.13022 Brewer, James W. (ed.) et al., Multiplicative ideal theory in commutative algebra. A tribute to the work of Robert Gilmer. New York, NY: Springer (ISBN 978-0-387-24600-0/hbk). 89-108 (2006). The authors present several interesting open questions dealing with various aspects of the theory of integer-valued polynomials. These questions concern, in particular, Bhargava factorials [M. Bhargava, J. Reine Angew. Math. 490, 101–127 (1997; Zbl 0899.13022)], and Newtonian domains, defined as domains \(D\) which contain a sequence \(a_n\) such that the polynomials \(f_n(X)=\prod_{k=0}^{n-1}(X-a_k)/(a_n-a_k)\) form a basis for the module of all polynomials over \(K\), the field of quotients of \(D\), which map \(D\) in \(D\).For the entire collection see [Zbl 1106.13001]. Reviewer: Władysław Narkiewicz (Wrocław) Cited in 2 ReviewsCited in 22 Documents MSC: 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 11B68 Bernoulli and Euler numbers and polynomials 11B83 Special sequences and polynomials 13B25 Polynomials over commutative rings 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations Keywords:integer-valued polynomials; factorial sequences; Newtonian domains Citations:Zbl 0899.13022 Software:OEIS × Cite Format Result Cite Review PDF Online Encyclopedia of Integer Sequences: a(n) = Product_{p prime} p^{ Sum_{k>=0} floor[(n-1)/((p-1)p^k)]}. a(n) = denominator(p(n, x)) / (n!*denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)*binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.