The Clausenian hypergeometric function ${}_3{F}_2$ with unit argument and negative integral parameter differences

New explicit as well as manifestly symmetric three-term summationformulas are derived for the Clausenian hypergeometric series ${}_3{F}_2(1)$ with negative integral parameter differences. Our results generalize and naturally extend several similar relations published, in recent years, by many authors. An appropriate and useful connection is established with the quite underestimated 1974 paper by P. W. Karlsson.