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 A006216 M1466 #39 Feb 17 2021 07:35:23
%S A006216 2,5,14,46,178,800,4094,23536,150178,1053440,8057774,66750976,
%T A006216 595380178,5688903680,57975175454,627692271616,7195247514178,
%U A006216 87056789995520,1108708685037134,14825405274259456,207676251991176178
%N A006216 Number of down-up permutations of n+4 starting with 4.
%C A006216 Entringer numbers.
%D A006216 R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
%D A006216 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006216 Vincenzo Librandi, <a href="/A006216/b006216.txt">Table of n, a(n) for n = 0..200</a>
%H A006216 B. Bauslaugh and F. Ruskey, <a href="http://webhome.cs.uvic.ca/~ruskey/Publications/AlternatingLex/AlternatingLex.html">Generating alternating permutations lexicographically</a>, Nordisk Tidskr. Informationsbehandling (BIT) 30 (1990) 16-26.
%H A006216 J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (<a href="http://neilsloane.com/doc/bous.txt">Abstract</a>, <a href="http://neilsloane.com/doc/bous.pdf">pdf</a>, <a href="http://neilsloane.com/doc/bous.ps">ps</a>).
%H A006216 C. Poupard, <a href="http://dx.doi.org/10.1016/0012-365X(82)90293-X">De nouvelles significations énumeratives des nombres d'Entringer</a>, Discrete Math., 38 (1982), 265-271.
%F A006216 a(n) = 3*E(n+2) - E(n), where E(j) = A000111(j) = j!*[x^j](sec(x) + tan(x)) are the up/down or Euler numbers. - _Emeric Deutsch_, May 15 2004
%F A006216 E.g.f.: 6/(cos(x)*(1 - sin(x))) - tan(x) - 4*sec(x). - _Sergei N. Gladkovskii_, Jun 04 2015
%F A006216 a(n) ~ 3*n^2 * 2^(n+4) * n! / Pi^(n+3). - _Vaclav Kotesovec_, Jun 04 2015
%e A006216 a(1) = 5 because we have 41325, 41523, 42314, 42513 and 43512.
%p A006216 f:=sec(x)+tan(x): fser:=series(f,x=0,30): E[0]:=1: for n from 1 to 25 do E[n]:=n!*coeff(fser,x^n) od: seq(3*E[n+2]-E[n],n=0..20);
%t A006216 e[0] = e[1] = 1; e[n_] := 2*Sum[ 4^m*Sum[ (i-(n-1)/2)^(n-1)*Binomial[ n-2*m-1, i-m]*(-1)^(n-i-1), {i, m, (n-1)/ 2}], {m, 0, (n-2)/2}]; a[0]=2; a[n_] := 3e[n+2] - e[n]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jan 27 2012, after _Emeric Deutsch_ *)
%o A006216 (PARI) {a(n) = local(v=[1], t); if( n<0, 0, for(k=2, n+4, t=0; v = vector(k, i, if( i>1, t += v[k+1-i]))); v[4])}; /* _Michael Somos_, Feb 03 2004 */
%Y A006216 Cf. A000111. Column k=3 in A008282.
%K A006216 nonn,easy,nice
%O A006216 0,1
%A A006216 _N. J. A. Sloane_