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%I #15 May 24 2016 16:34:55
%S 1,1,1,7,10,10,245,280,280,14700,15400,15400,1401400,1401400,1401400,
%T 196196000,190590400,190590400,38022784800,36212176000,36212176000,
%U 9759181432000,9161680528000,9161680528000,3206588184800000,2977546171600000,2977546171600000
%N a(n) is the number of abelian subgroups of maximal order in S_n.
%C The maximal order of an abelian subgroup in S_n is given by A000792.
%C Sequence becomes non-monotonic at n=20.
%F E.g.f.: exp(x^3/3!)*(1 + x^2/2 + 7*x^4/4!) + x - 1.
%F From _Benedict W. J. Irwin_, May 24 2016: (Start)
%F If n=1, a(n) = 1.
%F If n=2,5,8,11,..., a(n) = n!*2^(-(n+1)/3)*3^((2-n)/3)/Gamma((n+1)/3).
%F If n=3,6,9,12,..., a(n) = n!*6^(-n/3)/Gamma((n+3)/3).
%F If n=4,7,10,13,..., a(n) = n!*7*2^(-(n+5)/3)*3^((1-n)/3)/Gamma((n-1)/3).
%F (End)
%e a(4) = 7 because we have three cyclic groups: <(1234)> = <(1432)>, <(1243)> = <(1342)>, <(1324)> = <(1423)> and four groups isomorphhic to C_2 X C_2: <(12),(34)>, <(13),(24)>, <(14),(23)> , <(12)(34),(13)(24)> for a total of 7 distinct subgroups of maximal order 4.
%t nn = 25; Drop[Range[0, nn]! CoefficientList[Series[Exp[x^3/3!] (1 + x^2/2 + 7 x^4/4!) + x - 1, {x, 0, nn}], x], 1]
%Y Cf. A000792.
%K nonn,easy
%O 1,4
%A _Geoffrey Critzer_, Dec 06 2015
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