An analytic disproof of Robertson's conjecture (Q1173808)

Let the polynomials \(d_ n(x,y)=\sum_{i=0}^ n d_{ni}(y)x^ i\) be given by their generating function \[ \sum_{n=0}^ \infty d_ n(x,y)z^ n=\left({((1+z)/(1-z))^ x-1 \over 2xz}\right)^ y. \] It has been conjectured by Robertson that all the coefficients \(d_{ni}(1/2)\) are nonnegative. Fransén [Properties of Stirling polynomials, and a disproof of Robertson's conjecture, preprint] and \textit{F. W. Steutel} [J. Math. Anal. Appl. 158, No. 2, 578-582 (1991; Zbl 0731.41029)] disproved this conjecture by giving explicit counterexamples (e.g. \(d_{13,13}(1/2)<0\)). In this paper an indirect disproof is offered which uses manipulations of generating functions for Stirling polynomials and properties of Bernoulli numbers.