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%I #49 Feb 11 2020 18:23:59
%S 1,1,3,51,18731,408990251,921132763911411,324499299994016295527283,
%T 25190248259800264134073495741338539,
%U 576797123806621878513443912437627670334052360619
%N Number of chains of 2-element subsets of {0,1, 2, ..., n} that contain no consecutive integers.
%C For n >= 1, a(n) is the number of chains of binary reflexive symmetric matrices of order n.
%C The number of chains of strictly upper triangular or strictly lower triangular matrices.
%C Also, number of chains in power set of (n^2-n)/2 elements.
%C a(n) is the number of distinct reflexive symmetric fuzzy matrices of order n.
%H Alois P. Heinz, <a href="/A330302/b330302.txt">Table of n, a(n) for n = 0..29</a>
%H S. R. Kannan and Rajesh Kumar Mohapatra, <a href="https://arxiv.org/abs/1909.13678">Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques</a>, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
%H V. Murali, <a href="https://doi.org/10.1016/j.fss.2006.03.005">Combinatorics of counting finite fuzzy subsets</a>, Fuzzy Sets Syst., 157(17)(2006), 2403-2411.
%H V. Murali and B. Makamba, <a href="https://doi.org/10.1080/03081070512331318356">Finite Fuzzy Sets</a>, Int. J. Gen. Syst., Vol. 34 (1) (2005), pp. 61-75.
%H R. B. Nelsen and H. Schmidt, Jr., <a href="http://www.jstor.org/stable/2690450">Chains in power sets</a>, Math. Mag., 64 (1) (1991), 23-31.
%F a(n) = A007047((n^2-n)/2) = A007047(A161680(n)).
%p # P are the polynomials defined in A007047.
%p a:= n -> (m-> 2^m*subs(x=1/2, P(m, x)))(n*(n-1)/2):
%p seq(a(n), n=0..9);
%p # second Maple program:
%p b:= proc(n) option remember; `if`(n=0, 4,
%p add(b(n-j)*binomial(n, j), j=1..n))
%p end:
%p a:= n-> `if`(n<2, 1, b(n*(n-1)/2)-1):
%p seq(a(n), n=0..10); # _Alois P. Heinz_, Feb 11 2020
%t Array[2 PolyLog[-(#^2-#)/2, 1/2] - 1 &, 10, 0]
%t Table[2*PolyLog[-(n^2-n)/2, 1/2] - 1, {n, 0, 29}]
%t Table[LerchPhi[1/2, -(n^2-n)/2, 2]/2, {n, 0, 19}]
%Y Cf. A000079, A161680, A038719, A007047, A328044.
%K nonn
%O 0,3
%A S. R. Kannan, _Rajesh Kumar Mohapatra_, Jan 01 2020
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