A178749 -id:A178749 - cp's OEIS frontend

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

F. Chapoton, Jun 22 2010

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

			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;
   ...

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

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

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 *)

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)

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

A178738 Moebius inversion of a sequence related to powers of 2.

Original entry on oeis.org

1, -1, -1, 1, 2, -3, -5, 9, 15, -27, -49, 89, 164, -304, -565, 1057, 1987, -3745, -7085, 13445, 25575, -48771, -93210, 178481, 342392, -657935, -1266205, 2440323, 4709403, -9099507, -17602325, 34087058, 66076421, -128207979, -248983641

Offset: 1

F. Chapoton, Jun 08 2010

Only odd indices make sense. The given sequence is a(1), a(3), a(5), etc.
This should be related to the Coxeter transformations for the posets of diagonally symmetric paths in an n*n grid. - F. Chapoton, Jun 11 2010
Start from 1, 1, -2, -2, -4, -4, 8, 8, 16, 16, -32, -32, -64, -64, 128, ... which is A016116(n-1) with negative signs in blocks of 4, assuming offset 1. The Mobius transform of that sequence is b(n) = 1, 0, -3, -3, -5, -2, 7, 10, 18, 20, -33, -25, -65, -72, 135, 120, ... for n >= 1, and the current sequence is a(n) = b(2n-1)/(2n-1). - R. J. Mathar, Oct 29 2011

			b(1)=1*1; b(3)=-1*3; ...; b(9)=2*9.

Similar to A022553 and A131868

Also related to A178749. - F. Chapoton, Jun 11 2010

A := proc(n)
        (-1)^binomial(floor((n+1)/2),2) * 2^floor((n-1)/2) ;
end proc:
L := [seq(A(n),n=1..40)] ;
b := MOBIUS(L) ;
for i from 1 to nops(b) by 2 do
        printf("%d,", op(i,b)/i) ;
end do: # R. J. Mathar, Oct 29 2011

b[n_] := Sum[(-1)^Binomial[(d+1)/2, 2]*2^((d-1)/2)*MoebiusMu[n/d], {d, Divisors[n]}]/n;
a[n_] := b[2n - 1];
a /@ Range[35] (* Jean-François Alcover, Mar 16 2020 *)

def suite(n):
    return sum((-1)**binomial(((d+1)//2), 2) * 2**((d-1)//2) * moebius(n//d) for d in divisors(n)) // n
[suite(n) for n in range(1,22,2)]

I would like a more precise definition. - N. J. A. Sloane, Jun 08 2010

cp's OEIS Frontend

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

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