cp's OEIS Frontend

A006212 Number of down-up permutations of n+3 starting with n+1.

%I A006212 M3485 #44 Jul 08 2025 16:43:25
%S A006212 0,1,4,14,56,256,1324,7664,49136,345856,2652244,22014464,196658216,
%T A006212 1881389056,19192151164,207961585664,2385488163296,28879019769856,
%U A006212 367966308562084,4922409168011264,68978503204900376,1010472388453728256,15445185289163949004
%N A006212 Number of down-up permutations of n+3 starting with n+1.
%C A006212 Entringer numbers.
%D A006212 R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
%D A006212 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006212 Alois P. Heinz, <a href="/A006212/b006212.txt">Table of n, a(n) for n = 0..483</a>
%H A006212 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 A006212 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 A006212 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 A006212 From _Emeric Deutsch_, May 15 2004: (Start)
%F A006212 a(n) = Sum_{i=0..1+floor((n+1)/2)} (-1)^i * binomial(n, 2*i+1) * E[n+1-2i], where E[j] = A000111(j) = j!*[x^j](sec(x) + tan(x)) are the up/down or Euler numbers.
%F A006212 a(n) = T(n+2, n), where T is the triangle in A008282. (End)
%F A006212 a(n) = E[n+2] - E[n] where E[n] = A000111(n). - _Gerald McGarvey_, Oct 09 2006
%F A006212 E.g.f.: (sec(x) + tan(x))^2/cos(x) - (sec(x) + tan(x)). - _Sergei N. Gladkovskii_, Jun 29 2015
%F A006212 a(n) ~ n! * 2^(n+4) * n^2 / Pi^(n+3). - _Vaclav Kotesovec_, May 07 2020
%e A006212 a(2)=4 because we have 31425, 31524, 32415 and 32514.
%p A006212 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 A006212 # Alternatively after _Alois P. Heinz_ in A000111:
%p A006212 b := proc(u, o) option remember;
%p A006212 `if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
%p A006212 a := n -> b(n, 2): seq(a(n), n = 0..21); # _Peter Luschny_, Oct 27 2017
%t A006212 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 A006212 Cf. A000111, A008282.
%Y A006212 Column k=3 of A010094.
%K A006212 nonn,easy
%O A006212 0,3
%A A006212 _N. J. A. Sloane_
%E A006212 More terms from _Emeric Deutsch_, May 24 2004