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A218452
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Number of ways to factor (1 + x + x^2+ ... + x^(n - 1))^2 as the product of two monic polynomials of degree n - 1 with positive coefficients (counting order).
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0
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 7, 9, 9, 13, 11, 17, 19, 33
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OFFSET
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1,12
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COMMENTS
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a(n) is the number of ways one can divide the unit square in n possibly irregular lines * n possibly irregular columns (parallel to the sides) so that each of the diagonals of the n X n irregular checkerboard thus constructed has the same area as it would in a regular checkerboard. Alternatively, this is the number of ways to construct a pair of n-sided dice (probability distribution on the n sides, labeled 0 through n-1), no face having probability 0, so that the sum of the two dice follows the expected probability distribution for the sum of two fair n-sided dice. Note that a(n) is always odd because there is always the obvious factorization of (1+x+...+x^(n-1))^2 as 1+x+...+x^(n-1) times itself, and each other factorization counts twice.
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LINKS
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EXAMPLE
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For n=12 we have a(n)=3 because apart from the obvious factorization of (1+x+...+x^11)^2 as (1+x+...+x^11) times itself, there exist the factorizations p*q and q*p where p = (1-sqrt(3)*x+x^2) * (1-x+x^2) * (1+x^2) * (1+x+x^2)^2 * (1+x) and q = (1-sqrt(3)*x+x^2) * (1-x+x^2) * (1+x^2) * (1+sqrt(3)*x+x^2)^2 * (1+x), both of which have positive coefficients, and those are the only two possible.
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PROG
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(Sage)
R.<x> = AA['x']
def has_positive_coefficients(pol):
return not any(c <= 0 for c in pol.coeffs())
def trydie(m):
results = []
tmp = list(factor(sum([x^i for i in range(m)])))
facs = [f for (f, _) in tmp]
n = len(facs)
for i in range((3^n+1)//2):
exps = [(i//(3^k))%3 for k in range(n)]
coexps = [2-v for v in exps]
pol = R(prod([facs[k]^exps[k] for k in range(n)]))
copol = R(prod([facs[k]^coexps[k] for k in range(n)]))
if pol.degree()<m and copol.degree()<m and has_positive_coefficients(pol) and has_positive_coefficients(copol):
pol /= pol.subs({x:1})
copol /= copol.subs({x:1})
results += [(pol.coeffs(), copol.coeffs())]
return 2*len(results)-1
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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