A105586 -id:A105586 - cp's OEIS frontend

A105234 Central column of a Moebius-binomial triangle.

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

Offset: 0

Paul Barry, Apr 14 2005

Central column of A105586. Partial sums are A105235.

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

Cf. A105235, A105586.

Cf. A008683, A105597.

a[n_]:=Binomial[Abs[MoebiusMu[2n]],Abs[MoebiusMu[n]]];Table[a[n],{n,0,100}] (* James C. McMahon, Jan 22 2024 *)

A105234(n) = if(n<2,1,binomial(abs(moebius(2*n)), abs(moebius(n)))); \\ Antti Karttunen, Sep 13 2017

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

A105587 Row sums of a Möbius-binomial triangle.

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 7, 8, 2, 3, 11, 12, 4, 14, 15, 16, 5, 18, 6, 20, 7, 22, 23, 24, 8, 9, 27, 10, 11, 30, 31, 32, 12, 34, 35, 36, 13, 38, 39, 40, 14, 42, 43, 44, 15, 16, 47, 48, 17, 18, 19, 52, 20, 54, 21, 56, 22, 58, 59, 60, 23, 62, 63, 24, 25, 66, 67, 68, 26, 70, 71, 72, 27, 74, 75

Offset: 0

Paul Barry, Apr 14 2005

a(n) = n unless n is nonsquarefree, in which case a(n) = rank of n as a nonsquarefree number. Row sums of A105586.

			a(4) = 1 since 4 is the first nonsquarefree number;
a(8) = 2 since 8 is the second nonsquarefree number.

a(n) = Sum_{k=0..n} binomial(abs(mu(n)), abs(mu(k))).

A105588 Diagonal sums of a Moebius-binomial triangle.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 4, 3, 4, 4, 5, 5, 5, 5, 7, 7, 7, 7, 9, 8, 9, 8, 11, 9, 10, 9, 11, 10, 10, 10, 13, 12, 13, 12, 17, 14, 14, 14, 18, 16, 16, 17, 20, 17, 16, 16, 21, 18, 17, 17, 23, 20, 20, 18, 24, 22, 23, 21, 26, 23, 23, 22, 27, 24, 24, 24, 30, 25, 26, 24, 34, 27, 28, 26, 33, 28

Offset: 0

Paul Barry, Apr 14 2005

Diagonal sums of A105586.

Cf. A008683, A105586.

a(n) = Sum_{k=0..floor(n/2)} binomial(abs(mu(n-k)), abs(mu(k))).

A105589 Inverse of a Moebius-binomial triangle.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0

Offset: 0

Paul Barry, Apr 14 2005

Inverse of A105586. The inverse exists since binomial(abs(mu(n)),abs(mu(n)))=1 for all n.

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A105234 Central column of a Moebius-binomial triangle.

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A105587 Row sums of a Möbius-binomial triangle.

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A105588 Diagonal sums of a Moebius-binomial triangle.

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A105589 Inverse of a Moebius-binomial triangle.

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