  The modified Bernoulli numbers $B_{n}^{*}$ considered by Zagier are
generalized to modified N\"orlund polynomials ${B_{n}^{(\ell)*}}$. For
$\ell\in\mathbb{N}$, an explicit expression for the generating function for
these polynomials is obtained. Evaluations of some spectacular integrals
involving Chebyshev polynomials, and of a finite sum involving integrals of the
Hurwitz zeta function are also obtained. New results about the $\ell$-fold
convolution of the square hyperbolic secant distribution are obtained, such as
a differential-difference equation satisfied by a logarithmic moment and a
closed-form expression in terms of the Barnes zeta function.
