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 A006214 Number of down-up permutations of n+5 starting with n+1. (Formerly M3967) 0
 0, 5, 32, 178, 1024, 6320, 42272, 306448, 2401024, 20253440, 183194912, 1769901568, 18198049024, 198465167360, 2288729963552, 27831596812288, 355961301697024, 4777174607790080, 67129052143388192, 985743987073220608, 15098811288386497024, 240833888369219993600 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990. C. Poupard, De nouvelles significations énumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271. 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 (Abstract, pdf, ps). FORMULA 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 EXAMPLE a(1)=5 because we have 214365, 215364, 215463, 216354 and 216453. 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+3-2*i], i=0..floor((n-1)/2)): seq(a(n), n=0..16); # 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, 4): 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 + 4, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *) CROSSREFS Cf. A000111, A008282. Sequence in context: A272448 A193783 A271398 * A157704 A270565 A271165 Adjacent sequences:  A006211 A006212 A006213 * A006215 A006216 A006217 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Jean-François Alcover, Feb 12 2016 STATUS approved

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Last modified October 18 14:52 EDT 2019. Contains 328161 sequences. (Running on oeis4.)