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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

Table of n, a(n) for n=0..7.

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

Sequence in context: A079556 A232161 A052628 * A120417 A175185 A257610

Adjacent sequences:  A006850 A006851 A006852 * A006854 A006855 A006856

KEYWORD

nonn,nice,more

AUTHOR

N. J. A. Sloane

EXTENSIONS

a(7) from Sean A. Irvine, Aug 15 2017

STATUS

approved

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Last modified September 17 08:31 EDT 2019. Contains 327127 sequences. (Running on oeis4.)