A219613 E.g.f. tan(x/(1-x)).

0, 1, 2, 8, 48, 376, 3600, 40592, 525952, 7692928, 125303040, 2248366592, 44055035904, 935800603648, 21417131939840, 525346642337792, 13748654428323840, 382362034331877376, 11260657076602208256, 350082293087247269888, 11457214800338786713600

Offset: 0

Author

Geoffrey Critzer, Nov 23 2012

Keywords

• nonn

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.

Examples

			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.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 144

Crossrefs

Cf. A000182, A080832.

Programs

Mathematica

nn=21;Range[0,nn]!CoefficientList[Series[Tan[x/(1-x)],{x,0,nn}],x]

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