A178738 -id:A178738 - 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

A178749 n*a(n) provides the Moebius transform of signed central binomial coefficients.

Original entry on oeis.org

1, -1, -1, 1, 1, -1, -3, 4, 8, -13, -23, 39, 71, -121, -229, 400, 757, -1354, -2559, 4625, 8799, -16021, -30671, 56316, 108166, -200047, -385210, 716429, 1383331, -2585173, -5003791, 9391680, 18214565, -34318117, -66674463, 126044208, 245273927, -465067981

Offset: 1

F. Chapoton, Jun 09 2010

sign

This should be related to the Coxeter transformation for the Tamari lattices.

The source sequence is 1, -1, -2, 3, 6, -10, -20, 35, 70, -126, ... (A001405). Its Mobius transform is 1, -2, -3, 4, 5, -6, -21, 32, 72, -130, -253, 468, 923, ... and division of each term through n generates a(n). - R. J. Mathar, Jul 23 2012

			G.f. = x - x^2 - x^3 + x^4 + x^5 - x^6 - 3*x^7 + 4*x^8 + 8*x^9 - 13*x^10 + ...

Alois P. Heinz, Table of n, a(n) for n = 1..1000
F. Chapoton, Le module dendriforme sur le groupe cyclique, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2333-2350. In French.

Similar to A022553, A131868 and A178738.

Also related to A163210.

with(numtheory):
a:= n-> add(mobius(n/d)*[1$2, -1$2][1+irem(d, 4)]*
        binomial(d-1, iquo(d-1, 2)), d=divisors(n))/n:
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 05 2013

a[ n_] := If[ n < 1, 0, DivisorSum[ n, MoebiusMu[ n/#] (-1)^Quotient[ #, 2] Binomial[ # - 1, Quotient[ # - 1, 2]] &] / n]; (* Michael Somos, Sep 14 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, moebius(n/d) * (-1)^(d\2) * binomial(d-1, (d-1)\2)) / n)}; /* Michael Somos, Dec 23 2014 */

def lam(n):
    return (-1)**binomial(n, 2) * binomial(n - 1, (n - 1) // 2)
def a(n):
    return sum(moebius(n // d) * lam(d) for d in divisors(n)) // n
[a(n) for n in range(1, 20)]

A334125 Number of subsets of {1, 3, ..., 2n-1} which sum to 0 modulo 2n-1.

Original entry on oeis.org

2, 2, 2, 2, 4, 6, 10, 18, 30, 54, 98, 178, 328, 608, 1130, 2114, 3974, 7490, 14170, 26890, 51150, 97542, 186420, 356962, 684784, 1315870, 2532410, 4880646, 9418806, 18199014, 35204650, 68174116, 132152842, 256415958, 497967282, 967879954, 1882725390, 3665038872

Offset: 1

Jinyuan Wang, Apr 30 2020

nonn

			a(5) = 4 because there are 4 subsets of {1, 3, 5, 7, 9} which sum to 0 modulo 9: {}, {9}, {1, 3, 5}, {1, 3, 5, 9}.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A063776, A178738.

f:= proc(n) local V, k;
  V:= Vector(2*n-1);
  V[2*n-1]:= 1;
  for k from 1 to 2*n-1 by 2 do
    V:= V + V[[$(k+1)..(2*n-1),$1..k]]
  od;
  V[2*n-1]
end proc:
map(f, [$1..40]); # Robert Israel, May 12 2020

a(n) = {my(v=Vec(prod(i=1, n, x^(2*i-1)+1))); sum(i=0, n^2\(2*n-1), v[n^2+1-i*(2*n-1)]); }

If 2*k - 1 is a prime, then a(k) = (2^k - 2*(-1)^floor(k/2))/(2*k - 1).

Conjecture: a(n) = 2*abs(A178738(n)).

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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A178749 n*a(n) provides the Moebius transform of signed central binomial coefficients.

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Maple

Mathematica

PARI

Sage

A334125 Number of subsets of {1, 3, ..., 2n-1} which sum to 0 modulo 2n-1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Extensions

A178749 n*a(n) provides the Moebius transform of signed central binomial coefficients.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

A334125 Number of subsets of {1, 3, ..., 2*n-1} which sum to 0 modulo 2*n-1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A334125 Number of subsets of {1, 3, ..., 2n-1} which sum to 0 modulo 2n-1.