This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309845 #14 Aug 20 2019 02:51:10
%S A309845 1,2,1,4,5,32,61,544,1385,15872,50521,707584,2702765,44736512,
%T A309845 199360981,3807514624,19391512145,419730685952,2404879675441,
%U A309845 58177770225664,370371188237525,9902996106248192,69348874393137901,2030847773013704704,15514534163557086905,493842960380415967232
%N A309845 Expansion of e.g.f.: sec(x) + 2*tan(x).
%H A309845 F. Heneghan and T. K. Petersen, <a href="http://math.depaul.edu/tpeter21/MaxMinUpDownCMJ2v.pdf">Power series for up-down min-max permutations</a>, 2013.
%H A309845 Masato Kobayashi, <a href="https://arxiv.org/abs/1908.00701">A new refinement of Euler numbers on counting alternating permutations</a>, arXiv:1908.00701 [math.CO], 2019.
%F A309845 a(n-2) = |{up-down 2nd-max-lower permutations in S_n}| for n >= 2 (see Definition 3.4 in Kobayashi).
%F A309845 a(n) = A000111(n+2) - A164575(n) (See Definition 3.4 in Kobayashi).
%F A309845 a(n) = A225688(n) + A225689(n) - A164575(n) (See Heneghan-Petersen and Kobayashi articles).
%F A309845 a(2*n) = A000111(2*n) (See Lemma 3.8 in Kobayashi).
%F A309845 a(2*n+1) = 2*A000111(2*n+1) (See Lemma 3.8 in Kobayashi).
%t A309845 CoefficientList[Series[Sec[x]+2Tan[x],{x,0,25}],x]*Table[n!,{n,0,25}]
%o A309845 (PARI) my(x='x+O('x^30)); Vec(serlaplace(1/cos(x)+2*tan(x))) \\ _Michel Marcus_, Aug 20 2019
%Y A309845 Cf. A000111, A000182, A000364, A164575, A181937, A225688, A225689.
%K A309845 nonn,easy
%O A309845 0,2
%A A309845 _Stefano Spezia_, Aug 20 2019