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A178904
This should be related to the Coxeter transformations of the posets of partitions in rectangular boxes of size m times n.
5
1, -1, -1, 0, -1, 0, 0, 1, 1, 0, 0, -1, 1, -1, 0, 0, 1, -1, -1, 1, 0, 0, -1, 2, -3, 2, -1, 0, 0, 1, -3, 4, 4, -3, 1, 0, 0, -1, 3, -6, 8, -6, 3, -1, 0, 0, 1, -3, 9, -13, -13, 9, -3, 1, 0, 0, -1, 4, -11, 19, -23, 19, -11, 4, -1, 0, 0, 1, -5, 13, -27, 39, 39, -27, 13, -5, 1, 0, 0, -1, 5, -17, 38, -61, 71, -61, 38, -17, 5, -1, 0
OFFSET
0,24
COMMENTS
This table is symmetric: a(m,n)=a(n,m) for all m,n>=0.
EXAMPLE
a(0,0) = 1, a(1,0) = a(0,1) = -1.
Triangle begins:
1;
-1, -1;
0, -1, 0;
0, 1, 1, 0;
0, -1, 1, -1, 0;
0, 1, -1, -1, 1, 0;
0, -1, 2, -3, 2, -1, 0;
...
MATHEMATICA
b[m_, n_] := (-1)^Max[m, n]*Binomial[m+n, n]; A[m_, n_] := DivisorSum[ n+m+1, b[Floor[m/#], Floor[n/#]]*MoebiusMu[#]&]/(m+n+1); Table[A[m-n, n], {m, 0, 12}, {n, 0, m}] // Flatten (* Jean-François Alcover, Feb 23 2017, adapted from Python *)
PROG
(Sage)
def twisted_binomial(m, n):
return (-1)**max(m, n) * binomial(m + n, n)
def coefficients_A(m, n):
return sum(twisted_binomial(m // d, n // d) * moebius(d)
for d in divisors(m + n + 1)) / (m + n + 1)
matrix(ZZ, 8, 8, coefficients_A)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
F. Chapoton, Jun 22 2010
EXTENSIONS
Terms a(82) onward added by G. C. Greubel, Dec 10 2017
STATUS
approved