A178904 This should be related to the Coxeter transformations of the posets of partitions in rectangular boxes of size m times n.

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

Author

F. Chapoton, Jun 22 2010

Keywords

• sign
• tabl

Comments

This table is symmetric: a(m,n)=a(n,m) for all m,n>=0.

Examples

			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;
   ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Crossrefs

Cf. A178738, A178749, A022553, A131868, A163210.

Programs

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 *)

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)

Extensions

Terms a(82) onward added by G. C. Greubel, Dec 10 2017