%I A113896
%S A113896 1,4,864,2579890176
%N A113896 Number of maximal (and largest) superset towers with n base elements.
%C A113896 On 3 elements ABC, some tower (Halmos, "Naive Set Theory" among many) 
               that begins with the empty set can be written without loss of generality 
               as {0, A, AB, ABC}. But we need to have sets B, C, BC, AC included 
               somewhere too so that the thing is "largest", i.e., includes every 
               subset of {A,B,C}. For ABCD, there are 3^3 ways to include B,C,D 
               into AB,AC,AD,BC,BD,CD and 2^5 ways to include AC,AD,BC,BD,CD into 
               ABC,ABD,ACD,BCD. So a(4) = 3^3*2^5.
%F A113896 (n-1)^(n-1) * (n-2)^(nC2 - 1) * (n-3)^(nC3 - 1) *...* 2^(nC(n-2) - 1) 
               * 1^(n-1)
%e A113896 a(2) = 4 because:
%e A113896 (1) 0->A A->AB B->AB C->AB AB->ABC AC->ABC BC->ABC ABC->ABC maximal
%e A113896 (2) 0->A A->AB B->AB C->AC AB->ABC AC->ABC BC->ABC ABC->ABC maximal
%e A113896 (3) 0->A A->AB B->AC C->AB AB->ABC AC->ABC BC->ABC ABC->ABC maximal
%e A113896 (4) 0->A A->AB B->AC C->AC AB->ABC AC->ABC BC->ABC ABC->ABC maximal
%Y A113896 Sequence in context: A102195 A114766 A072725 this_sequence A159706 A006030 
               A087365
%Y A113896 Adjacent sequences: A113893 A113894 A113895 this_sequence A113897 A113898 
               A113899
%K A113896 nonn
%O A113896 0,2
%A A113896 Lee Corbin (lcorbin(AT)tsoft.com), Jan 28 2006