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A174725 a(n) = (A002033(n-1) + A008683(n))/2 5
1, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 4, 0, 4, 0, 4, 2, 2, 0, 10, 1, 2, 2, 4, 0, 6, 0, 8, 2, 2, 2, 13, 0, 2, 2, 10, 0, 6, 0, 4, 4, 2, 0, 24, 1, 4, 2, 4, 0, 10, 2, 10, 2, 2, 0, 22, 0, 2, 4, 16, 2, 6, 0, 4, 2, 6, 0, 38, 0, 2, 4, 4, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

From Mats Granvik, May 25 2017: (Start)

A002033(n-1) = a(n) + A174726(n).

A008683(n) = a(n) - A174726(n).

Let m = size of matrix a matrix T, and let T be defined as:

T(n,k) = if m = 1 then 1  else  if mod(n, k) = 0 then if and(n = k, n = m) then 0 else 1 else if and(n = 1, k = m) then 1 else 0

a(n) is then the number of permutation matrices with a positive contribution in the determinant of matrix T. The Determinant of T is equal to the Möbius function A008683, see Mathematica program below for how to compute the determinant.

A174726 is the number of permutation matrices with a negative contribution in the determinant of matrix T.

(End)

LINKS

Mats Granvik, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = (Mobius transform of a(n))+(Mobius transform of A174726). [Mats Granvik, Apr 04 2010]

From Mats Granvik, May 25 2017: (Start)

a(n) = Mobius transform of A002033.

a(n) = (A002033(n-1) + A008683(n))/2.

(End)

MATHEMATICA

(* From Mats Granvik, May 25 2017: (Start) *)

Clear[t, nn]; nn = 77; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, Sum[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Monitor[Table[Sum[If[Mod[n, k] == 0, MoebiusMu[k]*t[n/k, 1], 0], {k, 1, 77}], {n, 1, nn}], n]

(* The Möbius function as a determinant *) Table[Det[Table[Table[If[m == 1, 1, If[Mod[n, k] == 0, If[And[n == k, n == m], 0, 1], If[And[n == 1, k == m], 1, 0]]], {k, 1, m}], {n, 1, m}]], {m, 1, 42}]

(* (End) *)

CROSSREFS

Cf. A074206, A174726, A008683, A051731.

Sequence in context: A144765 A147588 A070824 * A071459 A070288 A165414

Adjacent sequences:  A174722 A174723 A174724 * A174726 A174727 A174728

KEYWORD

nonn

AUTHOR

Mats Granvik, Mar 28 2010

STATUS

approved

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Last modified June 22 17:50 EDT 2017. Contains 288633 sequences.