A006212 Number of down-up permutations of n+3 starting with n+1.

0, 1, 4, 14, 56, 256, 1324, 7664, 49136, 345856, 2652244, 22014464, 196658216, 1881389056, 19192151164, 207961585664, 2385488163296, 28879019769856, 367966308562084, 4922409168011264, 68978503204900376, 1010472388453728256, 15445185289163949004

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Entringer numbers.

Examples

			a(2)=4 because we have 31425, 31524, 32415 and 32514.
		

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.

Crossrefs

Cf. A000111, A008282.

Column k=3 of A010094.

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+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 *)

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

Extensions

More terms from Emeric Deutsch, May 24 2004