A006214 Number of down-up permutations of n+5 starting with n+1.
0, 5, 32, 178, 1024, 6320, 42272, 306448, 2401024, 20253440, 183194912, 1769901568, 18198049024, 198465167360, 2288729963552, 27831596812288, 355961301697024, 4777174607790080, 67129052143388192, 985743987073220608, 15098811288386497024, 240833888369219993600
Offset: 0
Examples
a(1)=5 because we have 214365, 215364, 215463, 216354 and 216453.
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 (1990), 16-26.
- 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 transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps).
Programs
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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
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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 *)
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
Extensions
More terms from Jean-François Alcover, Feb 12 2016
Comments