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A006213 Number of down-up permutations of n+4 starting with n+1.
(Formerly M1970)
0
0, 2, 10, 46, 224, 1202, 7120, 46366, 329984, 2551202, 21306880, 191252686, 1836652544, 18793429202, 204154071040, 2346705139006, 28459289083904, 363156549211202, 4864231397785600, 68237760828425326, 1000569392347480064, 15306487540377673202 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Entringer numbers.

REFERENCES

B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.

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.

J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon on transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).

C. Poupard, De nouvelles significations énumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.

FORMULA

a(n) = sum((-1)^i*binomial(n, 2i+1)*E[n+2-2i], i=0..1+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+3, n), where T is the triangle in A008282. - Emeric Deutsch, May 15 2004

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.

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

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Last modified January 20 14:40 EST 2019. Contains 319333 sequences. (Running on oeis4.)