Summary: Starting from a weight-system denoted by \(P\) and defined on the \(n\)-chord-diagrams with values in an arbitrary \(\mathbb{Q}\)-module, we give an explicit combinatorial formula for an invariant of \((n-1)\)-singular knots which has \(P\) as its derivative. The formula is defined for regular knot projections. Its invariance under singular Reidemeister moves is then proved.