A006853 Balanced colorings of n-cube. (Formerly M1299)

2, 2, 4, 18, 648, 3140062, 503483766022188, 171522187398423323340476473786538

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: A232161 A335684 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