Binomial self-inverse sequences and tangent coefficients

This paper treats the class of sequences {a_n} that satisfy the recurrence relation

$\text{a}{}_{\mathrm{2n+1}}\text{=∑}{}_{\mathrm{<mtext>k=0</mtext>}}{}^{\mathrm{<mtext>n</mtext>}}\text{(?1)}{}^{\mathrm{<mtext>k</mtext>}}\text{(}\text{n}\text{k}\text{a}{}_{\mathrm{k}}\text{d}{}^{\mathrm{n?k}}$

between the odd and even terms of {a_n} that involves the coefficients of tan(t), namely

$\text{a}{}_{\mathrm{2n+1}}\text{=∑}{}_{\mathrm{<mtext>k=0</mtext>}}{}^{\mathrm{<mtext>n</mtext>}}\text{(?1)}{}^{\mathrm{<mtext>k</mtext>}}\text{(}\text{2}\text{n}\text{+1}\text{2}\text{k}\text{+1}\text{)}\text{T}{}_{\mathrm{k}}\text{(d/2)}{}^{\mathrm{2k+1}}\text{a}{}_{\mathrm{2n?2k}}$

A combinatorial setting is then provided to elucidate the appearance of the tangent coefficients in this equation.