The tangent numbers $(T\_{2n+1})=(1,2,16,272,7936,...)$ (cf. OEIS: A000182) satisfy many recurrences. I would be interested to find references for the following which I think must be very old:
$T\_3 -2T\_1=0$, $T\_5 -8T\_3 =0,$ $T\_7 -18T\_5 +8T\_3 =0,...$ or more generally


$${T\_{2n + 1}} = \sum\limits\_{j \ge 1} {}{(-1)}^{j-1}{2^{2j}} {\binom{n+1}{2j}} {\frac{n+1-j}{n+1}}T\_{2n - 2j + 1}.$$



Reference request for an identity for tangent numbers
The following is too long for a comment, so let me type it as an answer though it does not literally answer your question.


Using the standard formula
 $$
T\\_{2k-1}=(-1)^{k-1}2^{2k}(2^{2k}-1)\frac{B\_{2k}}{2k},
 $$
your formula can be rewritten as
 $$
(2^{2n+2}-1)B\_{2n+2}=\sum\_{j\ge1}(-1)^{n-j+1}\binom{n+1}{2j}(2^{2n-2j+2}-1)B\_{2n-2j+2},
 $$
which looks "simpler", and also might be more recognisable by specialists, since identities for Bernoulli numbers are usually more "popular".


