%I A006215 M5007
%S A006215 0,16,122,800,5296,36976,275792,2204480,18870016,172585936,1681843712,
%T A006215 17411416160,190939611136,2211961358896,26999750469632,346419349043840
%N A006215 Number of down-up permutations of n+6 starting with n+1.
%C A006215 Entringer numbers.
%D A006215 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006215 R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli 
               numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
%D A006215 C. Poupard, De nouvelles significations enumeratives des nombres d'Entringer, 
               Discrete Math., 38 (1982), 265-271.
%H A006215 B. Bauslaugh and F. Ruskey, <a href="http://www.cs.uvic.ca/~fruskey/Publications/
               ">Generating alternating permutations lexicographically</a>, Nordisk 
               Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
%H A006215 J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: 
               the Boustrophedon on transform, J. Combin. Theory, 17A 44-54 1996 
               (<a href="http://www.research.att.com/~njas/doc/bous.txt">Abstract</
               a>, <a href="http://www.research.att.com/~njas/doc/bous.pdf">pdf</
               a>, <a href="http://www.research.att.com/~njas/doc/bous.ps">ps</a>
               ).
%H A006215 <a href="Sindx_To.html#tournament">Index entries for sequences related 
               to tournaments</a>
%F A006215 a(n)=sum((-1)^i*binomial(n, 2i+1)*E[n+4-2i], i=0..floor((n-1)/2)), where 
               E[j]=A000111(j)=j!*[x^j]((sec(x)+tan(x)) are the up/down or Euler 
               numbers. a(n)=T(n+5, n), where T is the triangle in A008282. - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), May 15 2004
%e A006215 a(1)=16 because we have 2143657, 2143756, 2153647, 2153746, 2154637, 
               2154736, 2163547, 2163745, 2164537, 2164735, 2165734, 2173546, 2173645, 
               2174536, 2174635 and 2175634.
%p A006215 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: a:=n->sum((-1)^i*binomial(n,2*i+1)*E[n+4-2*i],
               i=0..floor((n-1)/2)): seq(a(n),n=0..15);
%Y A006215 Cf. A000111, A008282.
%Y A006215 Sequence in context: A068880 A053883 A030508 this_sequence A060633 A125353 
               A126511
%Y A006215 Adjacent sequences: A006212 A006213 A006214 this_sequence A006216 A006217 
               A006218
%K A006215 nonn,easy
%O A006215 0,2
%A A006215 N. J. A. Sloane (njas(AT)research.att.com).