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 A006212 Number of down-up permutations of n+3 starting with n+1. (Formerly M3485) 3
 0, 1, 4, 14, 56, 256, 1324, 7664, 49136, 345856, 2652244, 22014464, 196658216, 1881389056, 19192151164, 207961585664, 2385488163296, 28879019769856, 367966308562084, 4922409168011264, 68978503204900376, 1010472388453728256, 15445185289163949004 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Entringer numbers. REFERENCES R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..483 B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 (1990), 16-26. 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 (Abstract, pdf, ps). C. Poupard, De nouvelles significations énumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271. FORMULA From Emeric Deutsch, May 15 2004: (Start) 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. a(n) = T(n+2, n), where T is the triangle in A008282. (End) a(n) = E[n+2] - E[n] where E[n] = A000111(n). - Gerald McGarvey, Oct 09 2006 E.g.f.: (sec(x) + tan(x))^2/cos(x) - (sec(x) + tan(x)). - Sergei N. Gladkovskii, Jun 29 2015 a(n) ~ n! * 2^(n+4) * n^2 / Pi^(n+3). - Vaclav Kotesovec, May 07 2020 EXAMPLE a(2)=4 because we have 31425, 31524, 32415 and 32514. MAPLE 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); # Alternatively after Alois P. Heinz in A000111: b := proc(u, o) option remember; `if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end: a := n -> b(n, 2): seq(a(n), n = 0..21); # Peter Luschny, Oct 27 2017 MATHEMATICA 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 *) CROSSREFS Cf. A000111, A008282. Column k=3 of A010094. Sequence in context: A149491 A073155 A346816 * A126701 A309514 A151884 Adjacent sequences:  A006209 A006210 A006211 * A006213 A006214 A006215 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Emeric Deutsch, May 24 2004 STATUS approved

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Last modified January 27 14:44 EST 2022. Contains 350607 sequences. (Running on oeis4.)