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[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
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
KEYWORD
nonn,nice,more
AUTHOR
EXTENSIONS
a(7) from Sean A. Irvine, Aug 15 2017
STATUS
approved