A006213 Number of down-up permutations of n+4 starting with n+1. (Formerly M1970)

0, 2, 10, 46, 224, 1202, 7120, 46366, 329984, 2551202, 21306880, 191252686, 1836652544, 18793429202, 204154071040, 2346705139006, 28459289083904, 363156549211202, 4864231397785600, 68237760828425326, 1000569392347480064, 15306487540377673202

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

Table of n, a(n) for n=0..21.

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.

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)

Example

a(1) = 2 because we have 21435 and 21534.

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

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 *)

Crossrefs

Cf. A000111, A008282.

Column k=4 of A010094.

Sequence in context: A204091 A221196 A137193 * A137635 A029706 A191644

Adjacent sequences: A006210 A006211 A006212 * A006214 A006215 A006216

Keyword

nonn,easy

Author

N. J. A. Sloane.

Extensions

More terms from Jean-François Alcover, Feb 12 2016

Status

approved