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%I M1299 #19 Aug 16 2017 05:28:01
%S 2,2,4,18,648,3140062,503483766022188,
%T 171522187398423323340476473786538
%N Balanced colorings of n-cube.
%C Number of ways to have center of gravity of an n-dimensional hypercube at center by weighting each vertex with 0 or 1.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H E. M. Palmer, R. C. Read and R. W. Robinson, <a href="https://doi.org/10.1023/A:1022487918212">Balancing the n-cube: a census of colorings</a>, J. Algebraic Combin., 1 (1992), 257-273.
%F For n > 0, a(n) = N(n,2^(n-1)) + 2 * Sum_{k=1..2^(n-1)-1} N(n, 2*k) where N(n,2k) = Sum_{(j)} N((j))^n * (-1)^c((j)) / h((j)) is the sum over all partitions (j) = (j[1],...,j[2k]) of 2*k (i.e., 2*k = Sum_{i=1..2*k} i*j[i]) and N((j)) is the coefficient of x^k in Product_{i=1..2*k} (1+x^i)^{j[i]}, c((j)) = Sum_{i} j[2*i], and h((j)) = Product_{i} j[i]! * i^{j[i]} [From Palmer et al.]. - _Sean A. Irvine_, Aug 15 2017
%e For a square (2 dimensions) there are 4 ways to weight each vertex with 0 or 1 while retaining center of gravity at center of the square, so a(2)=4.
%K nonn,nice,more
%O 0,1
%A _N. J. A. Sloane_
%E a(7) from _Sean A. Irvine_, Aug 15 2017
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