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A006215 Number of down-up permutations of n+6 starting with n+1.
(Formerly M5007)
1
0, 16, 122, 800, 5296, 36976, 275792, 2204480, 18870016, 172585936, 1681843712, 17411416160, 190939611136, 2211961358896, 26999750469632, 346419349043840, 4661658528710656, 65657186909139856, 966054350401175552, 14822897275566895520 (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..19.

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

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.

Index entries for sequences related to tournaments

FORMULA

a(n)=sum((-1)^i*binomial(n, 2i+1)*E[n+4-2i], 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+5, n), where T is the triangle in A008282. - Emeric Deutsch, May 15 2004

EXAMPLE

a(1)=16 because we have 2143657, 2143756, 2153647, 2153746, 2154637, 2154736, 2163547, 2163745, 2164537, 2164735, 2165734, 2173546, 2173645, 2174536, 2174635 and 2175634.

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+4-2*i], i=0..floor((n-1)/2)): seq(a(n), n=0..15);

# 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, 5): 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 + 5, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

CROSSREFS

Cf. A000111, A008282.

Sequence in context: A053883 A191484 A030508 * A227088 A060633 A125353

Adjacent sequences:  A006212 A006213 A006214 * A006216 A006217 A006218

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 18 20:57 EST 2019. Contains 319282 sequences. (Running on oeis4.)