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 A006853 Balanced colorings of n-cube. (Formerly M1299) 1
 2, 2, 4, 18, 648, 3140062, 503483766022188, 171522187398423323340476473786538 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Number of ways to have center of gravity of an n-dimensional hypercube at center by weighting each vertex with 0 or 1. REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS E. M. Palmer, R. C. Read and R. W. Robinson, Balancing the n-cube: a census of colorings, J. Algebraic Combin., 1 (1992), 257-273. FORMULA 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,...,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 EXAMPLE 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. CROSSREFS Sequence in context: A232161 A335684 A052628 * A120417 A175185 A257610 Adjacent sequences:  A006850 A006851 A006852 * A006854 A006855 A006856 KEYWORD nonn,nice,more AUTHOR EXTENSIONS a(7) from Sean A. Irvine, Aug 15 2017 STATUS approved

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Last modified May 12 10:48 EDT 2021. Contains 343821 sequences. (Running on oeis4.)