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A006214 Number of down-up permutations of n+5 starting with n+1.

%I A006214 M3967 #35 Jul 08 2025 16:43:34
%S A006214 0,5,32,178,1024,6320,42272,306448,2401024,20253440,183194912,
%T A006214 1769901568,18198049024,198465167360,2288729963552,27831596812288,
%U A006214 355961301697024,4777174607790080,67129052143388192,985743987073220608,15098811288386497024,240833888369219993600
%N A006214 Number of down-up permutations of n+5 starting with n+1.
%C A006214 Entringer numbers.
%D A006214 R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
%D A006214 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006214 B. Bauslaugh and F. Ruskey, <a href="http://dx.doi.org/10.1007/BF01932127">Generating alternating permutations lexicographically</a>, Nordisk Tidskr. Informationsbehandling (BIT) 30 (1990), 16-26.
%H A006214 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.
%H A006214 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>).
%F A006214 a(n) = sum((-1)^i*binomial(n, 2i+1)*E[n+3-2i], 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+4, n), where T is the triangle in A008282. - _Emeric Deutsch_, May 15 2004
%e A006214 a(1)=5 because we have 214365, 215364, 215463, 216354 and 216453.
%p A006214 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+3-2*i],i=0..floor((n-1)/2)): seq(a(n),n=0..16);
%p A006214 # Alternatively after _Alois P. Heinz_ in A000111:
%p A006214 b := proc(u, o) option remember;
%p A006214 `if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
%p A006214 a := n -> b(n, 4): seq(a(n), n = 0..21); # _Peter Luschny_, Oct 27 2017
%t A006214 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 + 4, n]; Array[a, 30, 0] (* _Jean-François Alcover_, Feb 12 2016 *)
%Y A006214 Cf. A000111, A008282.
%K A006214 nonn,easy
%O A006214 0,2
%A A006214 _N. J. A. Sloane_
%E A006214 More terms from _Jean-François Alcover_, Feb 12 2016