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A006212 Number of down-up permutations of n+3 starting with n+1.
(Formerly M3485)
2

%I M3485

%S 0,1,4,14,56,256,1324,7664,49136,345856,2652244,22014464,196658216,

%T 1881389056,19192151164,207961585664,2385488163296,28879019769856,

%U 367966308562084,4922409168011264,68978503204900376,1010472388453728256,15445185289163949004

%N Number of down-up permutations of n+3 starting with n+1.

%C Entringer numbers.

%D B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.

%D R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A006212/b006212.txt">Table of n, a(n) for n = 0..483</a>

%H 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://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 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 a(n) = sum((-1)^i*binomial(n, 2i+1)*E[n+1-2i], i=0..1+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+2, n), where T is the triangle in A008282. - _Emeric Deutsch_, May 15 2004

%F a(n) = E[n+2] - E[n] where E[n] = A000111(n). - _Gerald McGarvey_, Oct 09 2006

%F E.g.f.: (sec(x)+tan(x))^2/cos(x) - (sec(x)+tan(x)). - _Sergei N. Gladkovskii_, Jun 29 2015

%e a(2)=4 because we have 31425, 31524, 32415 and 32514.

%p 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+1-2*i],i=0..1+floor((n+1)/2)): seq(a(n),n=0..18);

%p # Alternatively after _Alois P. Heinz_ in A000111:

%p b := proc(u, o) option remember;

%p `if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:

%p a := n -> b(n, 2): seq(a(n), n = 0..21); # _Peter Luschny_, Oct 27 2017

%t 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 + 2, n]; Array[a, 30, 0] (* _Jean-François Alcover_, Feb 12 2016 *)

%Y Cf. A000111, A008282.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_.

%E More terms from _Emeric Deutsch_, May 24 2004

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Last modified November 19 08:53 EST 2018. Contains 317347 sequences. (Running on oeis4.)