A006214 Number of down-up permutations of n+5 starting with n+1. (Formerly M3967)

0, 5, 32, 178, 1024, 6320, 42272, 306448, 2401024, 20253440, 183194912, 1769901568, 18198049024, 198465167360, 2288729963552, 27831596812288, 355961301697024, 4777174607790080, 67129052143388192, 985743987073220608, 15098811288386497024, 240833888369219993600

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..21.

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

C. Poupard, De nouvelles significations énumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.

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).

Formula

a(n) = sum((-1)^i*binomial(n, 2i+1)*E[n+3-2i], 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+4, n), where T is the triangle in A008282. - Emeric Deutsch, May 15 2004

Example

a(1)=5 because we have 214365, 215364, 215463, 216354 and 216453.

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+3-2*i], i=0..floor((n-1)/2)): seq(a(n), n=0..16);
# 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, 4): 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 + 4, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

Crossrefs

Cf. A000111, A008282.

Sequence in context: A272448 A193783 A271398 * A157704 A270565 A271165

Adjacent sequences: A006211 A006212 A006213 * A006215 A006216 A006217

Keyword

nonn,easy

Author

N. J. A. Sloane.

Extensions

More terms from Jean-François Alcover, Feb 12 2016

Status

approved