Combinatorial sums,identities and trace identities of the 2 × 2 matrices

An asymptotic formula, involving integrals, is given for certain combinatorial sums. By evaluating a multi-integral it is then found that as n → ∞, the codimensions c_n(F₂) and the trace codimensions t_n(F₂) of F₂, the 2 × 2 matrices, are asymptotically equal: $\text{c}{}_{\mathrm{n}}\text{(F}{}_2\text{) ? t}{}_{\mathrm{n}}\text{(F}{}_2\text{)}$.