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A006216 Number of down-up permutations of n+4 starting with 4.
(Formerly M1466)
1
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; text; internal format)
OFFSET

0,1

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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.

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, May 15 2004

E.g.f.: 6/(cos(x)*(1-sin(x))) - tan(x) - 4*sec(x). - Sergei N. Gladkovskii, Jun 04 2015

a(n) ~ 3*n^2 * 2^(n+4) * n! / Pi^(n+3). - Vaclav Kotesovec, Jun 04 2015

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}] (* 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])}; /* Michael Somos, Feb 03 2004 */

CROSSREFS

Cf. A000111. Column k=3 in A008282.

Sequence in context: A275424 A107268 A231211 * A148337 A149899 A149900

Adjacent sequences:  A006213 A006214 A006215 * A006217 A006218 A006219

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 23 00:56 EST 2019. Contains 319365 sequences. (Running on oeis4.)