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A006212 Number of down-up permutations of n+3 starting with n+1.
(Formerly M3485)
2
0, 1, 4, 14, 56, 256, 1324, 7664, 49136, 345856, 2652244, 22014464, 196658216, 1881389056, 19192151164, 207961585664, 2385488163296, 28879019769856, 367966308562084, 4922409168011264, 68978503204900376, 1010472388453728256, 15445185289163949004 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

Alois P. Heinz, Table of n, a(n) for n = 0..483

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+1-2i], 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+2, n), where T is the triangle in A008282. - Emeric Deutsch, May 15 2004

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

EXAMPLE

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

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 *)

CROSSREFS

Cf. A000111, A008282.

Sequence in context: A259808 A149491 A073155 * A126701 A151884 A002735

Adjacent sequences:  A006209 A006210 A006211 * A006213 A006214 A006215

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Emeric Deutsch, May 24 2004

STATUS

approved

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Last modified October 19 17:55 EDT 2018. Contains 316376 sequences. (Running on oeis4.)