|
| |
|
|
A006216
|
|
Number of down-up permutations of n+4 starting with 4.
(Formerly M1466)
|
|
0
| |
|
|
2, 5, 14, 46, 178, 800, 4094, 23536, 150178, 1053440, 8057774, 66750976, 595380178, 5688903680, 57975175454, 627692271616, 7195247514178, 87056789995520, 1108708685037134, 14825405274259456, 207676251991176178
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,1
|
|
|
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.
C. Poupard, De nouvelles significations enumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| 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)=3E(n+2)-E(n), where E(j)=A000111(j)=j!*[x^j](sec(x)+tan(x)) are the up/down or Euler numbers. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 15 2004
|
|
|
EXAMPLE
| a(1)=5 because we have 41325,41523,42314,42513 and 43512.
|
|
|
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: seq(3*E[n+2]-E[n], n=0..20);
|
|
|
MATHEMATICA
| e[0] = e[1] = 1; e[n_] := 2*Sum[ 4^m*Sum[ (i-(n-1)/2)^(n-1)*Binomial[ n-2*m-1, i-m]*(-1)^(n-i-1), {i, m, (n-1)/ 2}], {m, 0, (n-2)/2}]; a[0]=2; a[n_] := 3e[n+2] - e[n]; Table[a[n], {n, 0, 20}] (* From Jean-François Alcover, Jan 27 2012, after Emeric Deutsch *)
|
|
|
PROG
| (PARI) a(n)=local(v=[1], t); if(n<0, 0, for(k=2, n+4, t=0; v=vector(k, i, if(i>1, t+=v[k+1-i]))); v[4]) (from Michael Somos)
|
|
|
CROSSREFS
| Cf. A000111. Column k=3 in A008282.
Sequence in context: A149898 A001680 A107268 * A148337 A149899 A149900
Adjacent sequences: A006213 A006214 A006215 * A006217 A006218 A006219
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
