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A006214 Number of down-up permutations of n+5 starting with n+1.
(Formerly M3967)
0
0, 5, 32, 178, 1024, 6320, 42272, 306448, 2401024, 20253440, 183194912, 1769901568, 18198049024, 198465167360, 2288729963552, 27831596812288, 355961301697024, 4777174607790080, 67129052143388192, 985743987073220608, 15098811288386497024, 240833888369219993600 (list; graph; refs; listen; history; text; internal format)
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 16-26 1990.

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

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).

FORMULA

a(n) = sum((-1)^i*binomial(n, 2i+1)*E[n+3-2i], i=0..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+4, n), where T is the triangle in A008282. - Emeric Deutsch, May 15 2004

EXAMPLE

a(1)=5 because we have 214365, 215364, 215463, 216354 and 216453.

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+3-2*i], i=0..floor((n-1)/2)): seq(a(n), n=0..16);

# 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, 4): 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 + 4, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

CROSSREFS

Cf. A000111, A008282.

Sequence in context: A272448 A193783 A271398 * A157704 A270565 A271165

Adjacent sequences:  A006211 A006212 A006213 * A006215 A006216 A006217

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 19 19:49 EST 2019. Contains 319309 sequences. (Running on oeis4.)