OFFSET
0,3
COMMENTS
Take each set partition of {1,2,...,n} into an odd number of blocks. Linearly order the elements within each block then form a "zag" permutation with the smallest element from each block. Here a "zag" permutation is a permutation a[1],a[2],...,a[k] such that a[1] < a[2] > a[3] < ... > a[k]. a(n) is the number of ways to order the blocks in accordance with each "zag" permutation.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 144
FORMULA
a(n) ~ 4/(Pi*(2+Pi))* n! * (1+2/Pi)^n. - Vaclav Kotesovec, Nov 25 2012
E.g.f.: x/(1-x)/T(0), where T(k) = 4*k+1 - x^2/((4*k+3)*(1-x)^2 - x^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 30 2013
EXAMPLE
a(3) = 8: The set partitions of {1,2,3} into an odd number of blocks are {1,2,3} and {1}{2}{3}. There are 6 ways to linearly order the elements of {1,2,3}. There are 2 such ways to order the blocks of the set partition {1}{2}{3}: {1}{3}{2} and {2}{3}{1}. 6+2=8.
MATHEMATICA
nn=21; Range[0, nn]!CoefficientList[Series[Tan[x/(1-x)], {x, 0, nn}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Nov 23 2012
STATUS
approved