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A006212 Number of down-up permutations of n+3 starting with n+1.
(Formerly M3485)
3
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
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
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+1-2i], 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. (End)
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
a(n) ~ n! * 2^(n+4) * n^2 / Pi^(n+3). - Vaclav Kotesovec, May 07 2020
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
Column k=3 of A010094.
Sequence in context: A073155 A365114 A346816 * A126701 A309514 A151884
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, May 24 2004
STATUS
approved

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)