An asymptotic series with Nörlund-number coefficients for higher-order Fubini numbers with a parameter

For real parameters r, α > 0, let F_n⁽ʳ⁾(α) denote the sequence built from Stirling numbers of the second kind by weighting with rising factorials of r and powers of α. Three familiar specializations sit inside this family: the Fubini numbers (r = α = 1), the higher-order Fubini numbers (α = 1, r a positive integer), and—after rescaling α in terms of p—the integer moments of a negative-binomial random variable. Our main theorem is an asymptotic expansion for F_n⁽ʳ⁾(α) in which the successive coefficients turn out to be the Nörlund (higher-order Bernoulli) numbers. A finite-termination phenomenon occurs at positive integer r: the algebraic singular expansion at the dominant singularity has exactly r nonzero terms, and the resulting r-term approximation agrees with F_n⁽ʳ⁾(α) to within an exponentially small relative error. We also prove a linear recurrence that holds for all real r, α > 0; it powers the numerical experiments at the end of the paper.