A080832 Expansion of e.g.f. exp(x) * (sec(exp(x) - 1))^2.

1, 1, 3, 13, 67, 421, 3115, 26349, 250867, 2655541, 30929019, 393019837, 5410699075, 80221867909, 1274393162827, 21594697199757, 388796268801427, 7411769447027413, 149143210226032923, 3159088788867736669

Offset: 0

Author

Emanuele Munarini, Mar 28 2003

Keywords

• easy
• nonn

Comments

Take the smallest element from each block of the set partitions of {1,2,...,n+1} into an odd number of blocks. Form a "zag" 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 such "zag" permutations. - Geoffrey Critzer, Nov 23 2012

Links

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

Crossrefs

Cf. A000182, A219613.

Programs

Maple

seq(coeff(series(factorial(n)*exp(x)*(sec(exp(x)-1))^2, x,n+1),x,n),n=0..25); # Muniru A Asiru, Jul 28 2018

Mathematica

nn=21;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[Tan[Exp[x]-1],{x,0,nn}],x],1]  (* Geoffrey Critzer, Nov 23 2012 *)

Formula

E.g.f.: exp(x) / (cos(exp(x) - 1))^2.

The sequence 0, 1, 1, 3, ... has e.g.f. tan(exp(x)-1). It has general term sum{k=0..n, S2(n, k) A009006(k)} for n>1 (S2(n, k) Stirling numbers of second kind). - Paul Barry, Apr 20 2005

a(n) ~ 2*n * n! / ((2+Pi) * (log(1+Pi/2))^(n+2)). - Vaclav Kotesovec, Jul 28 2018