Barry, Paul Combinatorial polynomials as moments, Hankel transforms, and exponential Riordan arrays. (English) Zbl 1241.11024 J. Integer Seq. 14, No. 6, Article 11.6.7, 14 p. (2011). It is shown in the case of two combinatorial polynomials that they can exhibited as moments of parametrized families of orthogonal polynomials, and hence derive their Hankel transforms. Exponentials Riordan arrays are the main vehicles used for this. Reviewer: Cristinel Mortici (Targoviste) Cited in 1 Document MSC: 11B83 Special sequences and polynomials 05A15 Exact enumeration problems, generating functions 11C20 Matrices, determinants in number theory 15B05 Toeplitz, Cauchy, and related matrices 15B36 Matrices of integers 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis Keywords:integer sequence; exponential Riordan array; Touchard polynomial; exponential polynomial; moments; orthogonal polynomials; Hankel determinant; Hankel transform Software:OEIS PDFBibTeX XMLCite \textit{P. Barry}, J. Integer Seq. 14, No. 6, Article 11.6.7, 14 p. (2011; Zbl 1241.11024) Full Text: arXiv EMIS Online Encyclopedia of Integer Sequences: Number of ”sets of lists”: number of partitions of {1,...,n} into any number of lists, where a list means an ordered subset. Pascal’s triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0 <= k <= n. Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n. Triangle of Eulerian numbers T(n,k) (n >= 1, 1 <= k <= n) read by rows. Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x). Triangle of permutation coefficients arranged with 1’s on the diagonal. Also, triangle of permutations on n letters with exactly k+1 cycles and with the first k+1 letters in separate cycles. Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n. The matrix inverse of the unsigned Lah numbers A271703. E.g.f.: exp(x/(1+x)).