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A006213 Number of down-up permutations of n+4 starting with n+1.

%I A006213 M1970 #43 Jul 08 2025 16:43:30
%S A006213 0,2,10,46,224,1202,7120,46366,329984,2551202,21306880,191252686,
%T A006213 1836652544,18793429202,204154071040,2346705139006,28459289083904,
%U A006213 363156549211202,4864231397785600,68237760828425326,1000569392347480064,15306487540377673202
%N A006213 Number of down-up permutations of n+4 starting with n+1.
%C A006213 Entringer numbers.
%D A006213 R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
%D A006213 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006213 B. Bauslaugh and F. Ruskey, <a href="https://doi.org/10.1007/BF01932127">Generating alternating permutations lexicographically</a>, Nordisk Tidskr. Informationsbehandling (BIT) 30 (1990), 16-26.
%H A006213 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 A006213 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 A006213 From _Emeric Deutsch_, May 15 2004: (Start)
%F A006213 a(n) = Sum_{i=0..1+floor((n+1)/2)} (-1)^i * binomial(n, 2*i+1) * E[n+2-2i], where E[j] = A000111(j) = j!*[x^j](sec(x) + tan(x)) are the up/down or Euler numbers.
%F A006213 a(n) = T(n+3, n), where T is the triangle in A008282. (End)
%e A006213 a(1) = 2 because we have 21435 and 21534.
%p A006213 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+2-2*i],i=0..1+floor((n+1)/2)): seq(a(n),n=0..17);
%p A006213 # Alternatively after _Alois P. Heinz_ in A000111:
%p A006213 b := proc(u, o) option remember;
%p A006213 `if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
%p A006213 a := n -> b(n, 3): seq(a(n), n = 0..21); # _Peter Luschny_, Oct 27 2017
%t A006213 t[n_, 0] := If[n == 0, 1, 0]; t[n_ , k_ ] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n + 3, n]; Array[a, 30, 0] (* _Jean-François Alcover_, Feb 12 2016 *)
%Y A006213 Cf. A000111, A008282.
%Y A006213 Column k=4 of A010094.
%K A006213 nonn,easy
%O A006213 0,2
%A A006213 _N. J. A. Sloane_
%E A006213 More terms from _Jean-François Alcover_, Feb 12 2016