A006215 Number of down-up permutations of n+6 starting with n+1.

0, 16, 122, 800, 5296, 36976, 275792, 2204480, 18870016, 172585936, 1681843712, 17411416160, 190939611136, 2211961358896, 26999750469632, 346419349043840, 4661658528710656, 65657186909139856, 966054350401175552, 14822897275566895520

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Entringer numbers.

Examples

			a(1)=16 because we have 2143657, 2143756, 2153647, 2153746, 2154637, 2154736, 2163547, 2163745, 2164537, 2164735, 2165734, 2173546, 2173645, 2174536, 2174635 and 2175634.
		

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.
• 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.
• Index entries for sequences related to tournaments

Crossrefs

Cf. A000111, A008282.

Programs

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+4-2*i],i=0..floor((n-1)/2)): seq(a(n),n=0..15);
# 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, 5): 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 + 5, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

Formula

From Emeric Deutsch, May 15 2004: (Start)

a(n) = Sum_{i=0..floor((n-1)/2)} (-1)^i * binomial(n, 2*i+1) * E[n + 4 - 2*i], where E[j] = A000111(j) = j! * [x^j] (sec(x) + tan(x)) are the up/down or Euler numbers.

a(n) = T(n+5, n), where T is the triangle in A008282. (End)

Extensions

More terms from Jean-François Alcover, Feb 12 2016