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Search: id:A006212
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| A006212 |
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Number of down-up permutations of n+3 starting with n+1.
(Formerly M3485)
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+0
1
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| 0, 1, 4, 14, 56, 256, 1324, 7664, 49136, 345856, 2652244, 22014464, 196658216, 1881389056, 19192151164, 207961585664, 2385488163296, 28879019769856, 367966308562084, 4922409168011264, 68978503204900376, 1010472388453728256
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Entringer numbers.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
C. Poupard, De nouvelles significations enumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.
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LINKS
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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).
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FORMULA
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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 (deutsch(AT)duke.poly.edu), May 15 2004
a(n) = E[n+2] - E[n] where E[n] = A000111(n). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 09 2006
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EXAMPLE
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a(2)=4 because we have 31425, 31524, 32415 and 32514.
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MAPLE
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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);
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CROSSREFS
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Cf. A000111, A008282.
Sequence in context: A132837 A149491 A073155 this_sequence A126701 A151884 A002735
Adjacent sequences: A006209 A006210 A006211 this_sequence A006213 A006214 A006215
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 24 2004
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