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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.

%I A178904 #16 Mar 16 2020 14:25:59
%S A178904 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,
%T A178904 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,
%U A178904 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
%N A178904 This should be related to the Coxeter transformations of the posets of partitions in rectangular boxes of size m times n.
%C A178904 This table is symmetric: a(m,n)=a(n,m) for all m,n>=0.
%H A178904 G. C. Greubel, <a href="/A178904/b178904.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%e A178904 a(0,0) = 1, a(1,0) = a(0,1) = -1.
%e A178904 Triangle begins:
%e A178904    1;
%e A178904   -1, -1;
%e A178904    0, -1,  0;
%e A178904    0,  1,  1,  0;
%e A178904    0, -1,  1, -1,  0;
%e A178904    0,  1, -1, -1,  1,  0;
%e A178904    0, -1,  2, -3,  2, -1, 0;
%e A178904    ...
%t A178904 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 *)
%o A178904 (Sage)
%o A178904 def twisted_binomial(m, n):
%o A178904     return (-1)**max(m, n) * binomial(m + n, n)
%o A178904 def coefficients_A(m, n):
%o A178904     return sum(twisted_binomial(m // d, n // d) * moebius(d)
%o A178904            for d in divisors(m + n + 1)) / (m + n + 1)
%o A178904 matrix(ZZ, 8, 8, coefficients_A)
%Y A178904 Cf. A178738, A178749, A022553, A131868, A163210.
%K A178904 sign,tabl
%O A178904 0,24
%A A178904 _F. Chapoton_, Jun 22 2010
%E A178904 Terms a(82) onward added by _G. C. Greubel_, Dec 10 2017