A178963 E.g.f.: (3+2*sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2))/(exp(-x)+2*exp(x/2)*cos(sqrt(3)*x/2)).

1, 1, 1, 1, 3, 9, 19, 99, 477, 1513, 11259, 74601, 315523, 3052323, 25740261, 136085041, 1620265923, 16591655817, 105261234643, 1488257158851, 17929265150637, 132705221399353, 2172534146099019, 30098784753112329, 254604707462013571, 4736552519729393091, 74180579084559895221, 705927677520644167681, 14708695606607601165843, 256937013876000351610089, 2716778010767155313771539

Offset

0,5

Comments

According to Mendes and Remmel, p. 56, this is the e.g.f. for 3-alternating permutations.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

J. M. Luck, On the frequencies of patterns of rises and falls, arXiv preprint arXiv:1309.7764 [cond-mat.stat-mech], 2013-2014.

Peter Luschny, An old operation on sequences: the Seidel transform.

Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page.

Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]

Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]

Formula

a(3*n) = A002115(n). - Peter Luschny, Aug 02 2012

Maple

A178963_list := proc(dim) local E, DIM, n, k;
DIM := dim-1; E := array(0..DIM, 0..DIM); E[0, 0] := 1;
for n from 1 to DIM do
if n mod 3 = 0 then E[n, 0] := 0 ;
   for k from n-1 by -1 to 0 do E[k, n-k] := E[k+1, n-k-1] + E[k, n-k-1] od;
else E[0, n] := 0;
   for k from 1 by 1 to n do E[k, n-k] := E[k-1, n-k+1] + E[k-1, n-k] od;
fi od; [E[0, 0], seq(E[k, 0]+E[0, k], k=1..DIM)] end:
A178963_list(30);  # Peter Luschny, Apr 02 2012
# Alternatively, using a bivariate exponential generating function:
A178963 := proc(n) local g, p, q;
g := (x, z) -> 3*exp(x*z)/(exp(z)+2*exp(-z/2)*cos(z*sqrt(3)/2));
p := (n, x) -> n!*coeff(series(g(x, z), z, n+2), z, n);
q := (n, m) -> if modp(n, m) = 0 then 0 else 1 fi:
(-1)^floor(n/3)*p(n, q(n, 3)) end:
seq(A178963(i), i=0..30); # Peter Luschny, Jun 06 2012
# third Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
     `if`(t=0, add(b(u-j, o+j-1, irem(t+1, 3)), j=1..u),
               add(b(u+j-1, o-j, irem(t+1, 3)), j=1..o)))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 29 2014

Mathematica

max = 30; f[x_] := (E^x*(2*Sqrt[3]*E^(x/2)*Sin[(Sqrt[3]*x)/2] + 3))/(2*E^((3*x)/2)*Cos[(Sqrt[3]*x)/2] + 1); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! // Simplify (* Jean-François Alcover, Sep 16 2013 *)

Prog

(Sage) # uses[A from A181936]
A178963 = lambda n: (-1)^int(is_odd(n//3))*A(3, n)
print([A178963(n) for n in (0..30)]) # Peter Luschny, Jan 24 2017

Crossrefs

Number of m-alternating permutations: A000012 (m=1), A000111 (m=2), A178963 (m=3), A178964 (m=4), A181936 (m=5).

Cf. A181937, A002115.

Cf. A249402, A249583 (alternative definitions of 3-alternating permutations).

Sequence in context: A146066 A164283 A375231 * A033315 A200612 A355136

Adjacent sequences: A178960 A178961 A178962 * A178964 A178965 A178966

Keyword

nonn

Author

N. J. A. Sloane, Dec 31 2010

Status

approved