A105595 -id:A105595 - cp's OEIS frontend

A105596 a(n) = Sum_{k=0..n} A105595(k)(-1)^kA105595(n-k) (interpolated zeros suppressed).

1, 5, 13, 17, 25, 25, 33, 21, 9, -15, -23, -3, -11, -31, -47, -35, 5, -47, -83, -75, -211, -295, -267, -267, -99, -107, -159, -415, -347, -679, -279, -583, -395, -839, -1031, -1291, -1139, -1883, -1519, -1643, -855, -1591, -1571, -1851, -1195, -2419, -1923, -2179, -891, -1919, -2535

Offset: 0

Paul Barry, Apr 14 2005

Conjecture : a(2n)=1 mod 4 for all n, a(2n+1)=0 for all n.

Cf. A105594, A105595.

A105596 := proc(n)
    add(A105595(k)*(-1)^k*A105595(2*n-k),k=0..2*n) ;
end proc:
seq(A105596(n),n=0..50) ; # R. J. Mathar, Nov 28 2014

A105594[n_, k_] := A105594[n, k] = Sum[Abs[MoebiusMu[ Binomial[n, j]]*Mod[Binomial[j, k], 2]], {j, 0, n}]//Mod[#, 2]&;
A105595[n_] := Sum[A105594[n, k], {k, 0, n}];
a[n_] := Sum[A105595[k]*(-1)^k*A105595[2n - k], {k, 0, 2n}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 01 2023, after R. J. Mathar *)

A105594 Triangle read by rows: abs(A103447)*A047999 mod 2.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1

Offset: 0

Paul Barry, Apr 14 2005

Row sums are A105595.

			Triangle starts
1;
0,1;
1,1,1;
0,0,0,1;
1,0,1,0,1;
0,1,0,1,0,1;
0,0,0,0,1,1,1;

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)

Cf. A047999, A103447, A105595, A105596.

A105594 := proc(n,k)
    add( abs(numtheory[mobius](binomial(n,j)))*modp(binomial(j,k),2) ,j=0..n) ;
    % mod 2 ;
end proc: # R. J. Mathar, Nov 28 2014

T[n_, k_] := Sum[Abs[MoebiusMu[Binomial[n, j]]*Mod[Binomial[j, k], 2]], {j, 0, n}] // Mod[#, 2]&;
Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 15 2020 *)

T(n, k) = mod(Sum_{j=0..n}(abs(mu(binomial(n,j)))*mod(binomial(j,k),2)), 2).

A105597 Central numbers in a Moebius-binomial triangle.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Paul Barry, Apr 14 2005

Central numbers in A105595.
There seems to be a typo in above comment, maybe A105594 was intended? - Antti Karttunen, Aug 27 2017
Partial sums are A105598.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for characteristic functions

a[n_]:= Binomial[Abs[MoebiusMu[n]],Abs[MoebiusMu[Floor[n/2]]]];Table[a[n],{n,0,100}] (* James C. McMahon, Jan 23 2024 *)

A105597(n) = if(n<2,1,binomial(abs(moebius(n)), abs(moebius(n\2)))); \\ Antti Karttunen, Aug 27 2017

a(n) = binomial(abs(mu(n)), abs(mu(floor(n/2)))).

cp's OEIS Frontend

A105596 a(n) = Sum_{k=0..n} A105595(k)(-1)^kA105595(n-k) (interpolated zeros suppressed).

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A105594 Triangle read by rows: abs(A103447)*A047999 mod 2.

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A105597 Central numbers in a Moebius-binomial triangle.

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cp's OEIS Frontend

A105596 a(n) = Sum_{k=0..n} A105595(k)*(-1)^k*A105595(n-k) (interpolated zeros suppressed).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

A105594 Triangle read by rows: abs(A103447)*A047999 mod 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A105597 Central numbers in a Moebius-binomial triangle.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula

A105596 a(n) = Sum_{k=0..n} A105595(k)(-1)^kA105595(n-k) (interpolated zeros suppressed).