%I A002135 M1513 N0594
%S A002135 1,1,2,5,17,73,388,2461,18155,152531,1436714,14986879,171453343,
%T A002135 2134070335,28708008128,415017867707,6416208498137,105630583492969,
%U A002135 1844908072865290,34071573484225549,663368639907213281,13580208904207073801
%N A002135 Number of terms in a symmetrical determinant: a(n) = n.a(n-1) - (n-1)(n-2)a(n-3)/
               2.
%C A002135 a(n) is the number of collections of necklaces created by using exactly 
               n different colored beads (to make the entire collection). [From 
               Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Apr 19 2009]
%D A002135 A. C. Aitken, On the number of distinct terms in the expansion of symmetric 
               and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5.
%D A002135 A. Cayley, On the number of distinct terms in a symmetrical or partially 
               symmetrical determinant, Collected Mathematical Papers. Vols. 1-13, 
               Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 190.
%D A002135 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #12, a_n.
%D A002135 P. A. MacMahon, Combinations derived from m identical sets of n different 
               letters and their connexion with general magic squares, Proc. London 
               Math. Soc., 17 (1917), 25-41.
%D A002135 Problem E2297, Amer. Math. Monthly, 79 (1972), 519-520.
%D A002135 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002135 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002135 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Example 5.2.9 and Problem 5.22.
%H A002135 T. D. Noe, <a href="b002135.txt">Table of n, a(n) for n=0..100</a>
%F A002135 E.g.f.: (1-x)^(-1/2)*exp(x/2+x^2/4). a(n+1) = (n+1)*a(n) - binomial(n, 
               2)*a(n-2) - Comtet.
%F A002135 Asymptotics: a(n):=sqrt(2) exp(3/4 -n ) n^n (1+O(1/n)) [From Pietro Majer 
               (majer(AT)dm.unipi.it), Oct 27 2009]
%Y A002135 Cf. A059422, A059423, A059424.
%Y A002135 Sequence in context: A007779 A084161 A102038 this_sequence A007868 A136726 
               A112831
%Y A002135 Adjacent sequences: A002132 A002133 A002134 this_sequence A002136 A002137 
               A002138
%K A002135 nonn,nice,easy
%O A002135 0,3
%A A002135 N. J. A. Sloane (njas(AT)research.att.com).