OFFSET
0,3
COMMENTS
Number of cyclically alternating permutations of length 2n. Example: a(2)=4 because we have 1324, 1423, 2314, and 2413 (3412 is alternating but not cyclically alternating).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001-2003.
N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
FORMULA
a(n) = 2^(n-1)*(2^n-1)*|B_n|.
E.g.f.: tan(x)*x/2 (even part).
a(n) = (2*n)!*Pi^(-2*n)*(4^n-1)*Li{2*n}(1) for n > 0. - Peter Luschny, Jun 29 2012
G.f.: Q(0)*x/(1-4*x), where Q(k) = 1 - 16*x^2*(k+2)*(k+1)^3/( 16*x^2*(k+2)*(k+1)^3 - (1 - 8*x*k^2 - 12*x*k -4*x)*(1 - 8*x*k^2 - 28*x*k -24*x)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 23 2013
a(n) = A009752(n)/2. - Alois P. Heinz, Aug 17 2021
a(n) = (-1)^n*2*n*PolyLog(1 - 2*n, -i). - Peter Luschny, Aug 17 2021
MAPLE
a := n -> (-1)^n*2^(2*n-1)*(1-2^(2*n))*bernoulli(2*n); # Peter Luschny, Jun 08 2009
MATHEMATICA
nn = 30; t = Range[0, nn]! CoefficientList[Series[Tan[x]*x/2, {x, 0, nn}], x]; Take[t, {1, nn, 2}]
Table[(-1)^n 2 n PolyLog[1 - 2 n, -I], {n, 0, 19}] (* Peter Luschny, Aug 17 2021 *)
PROG
(Python)
from itertools import accumulate, islice, count
def A024255_gen(): # generator of terms
yield from (0, 1)
blist = (0, 1)
for n in count(2):
yield n*(blist := tuple(accumulate(reversed(tuple(accumulate(reversed(blist), initial=0))), initial=0)))[-1]
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by Emeric Deutsch, Jul 01 2009
STATUS
approved