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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)). 8
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 (list; graph; refs; listen; history; text; internal format)
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, 2013

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

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)

# Function A(m, n) defined in 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: A147146 A146066 A164283 * A033315 A200612 A073716

Adjacent sequences:  A178960 A178961 A178962 * A178964 A178965 A178966

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 31 2010

STATUS

approved

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Last modified November 17 20:39 EST 2018. Contains 317278 sequences. (Running on oeis4.)