Obtain closed-form coefficient formulas for multidegrees of generalized barbell graphs

Develop simplified closed-form expressions for the coefficients of the multidegree polynomials of generalized barbell graphs formed by connecting more than two complete graphs (possibly of different sizes) by single edges, beyond the product-of-polynomials expressions obtainable from the multidegree formula for J_G.

Background

For standard barbell graphs B_n (two cliques joined by an edge), the authors derived explicit multidegree formulas and observed striking coefficient patterns (consecutive odd numbers). They suggest natural generalizations to multiple ‘bells’ and to bells of different sizes.

Although their methods yield multidegrees for these generalizations as products of simpler polynomials, they were unable to simplify these products into explicit coefficient formulas, indicating a concrete gap between computable product forms and closed-form coefficient expressions.