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%I #16 Nov 10 2013 11:32:38
%S 1,0,3,75,3108,205125,19839633,2643131400,463873573803,
%T 103710628476075,28775903316814668,9702563010998171325,
%U 3907429085273025561153,1852516229654506870381200,1021325008815288529961197683,647900078249178232882473232875
%N E.g.f.: sec(sec(x)-1) (even-indexed coefficients only).
%C Call a zig permutation a permutation p(1),p(2),...,p(2n) such that p(1)>p(2)< ... > p(2n) Cf. A000364. Consider the set of all set partitions of {1,2,...,2n} into an even number of even sized blocks. a(n) is the number of ways to build a zig permutation on each block and then build a zig permutation on the set formed from a representative (say the smallest element) of each block.
%H Alois P. Heinz, <a href="/A228841/b228841.txt">Table of n, a(n) for n = 0..200</a>
%e a(3) = 75. There are 15 set partitions of {1,2,3,4,5,6} that have an even number of even sized blocks Cf. A059386. They all have the same structure: 2/4. We build a zig permutation on each block in 1*5=5 ways. For each of these we then build a zig permutation on a representative from each of the 2 blocks in 1 way. So 5*1=5 and there are 15 such partitions so 5 *15 =75.
%t nn=30;Insert[Select[Range[0,nn]!CoefficientList[Series[Sec[Sec[x]-1],{x,0,nn}],x],#>0&],0,2]
%Y Cf. A219613, A080832, A003718.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Nov 10 2013
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