A006213 Number of down-up permutations of n+4 starting with n+1.
0, 2, 10, 46, 224, 1202, 7120, 46366, 329984, 2551202, 21306880, 191252686, 1836652544, 18793429202, 204154071040, 2346705139006, 28459289083904, 363156549211202, 4864231397785600, 68237760828425326, 1000569392347480064, 15306487540377673202
Offset: 0
Examples
a(1) = 2 because we have 21435 and 21534.
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.
- 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.
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+2-2*i],i=0..1+floor((n+1)/2)): seq(a(n),n=0..17); # 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, 3): 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 + 3, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)
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+2-2i], where E[j] = A000111(j) = j!*[x^j](sec(x) + tan(x)) are the up/down or Euler numbers.
a(n) = T(n+3, n), where T is the triangle in A008282. (End)
Extensions
More terms from Jean-François Alcover, Feb 12 2016
Comments