A085542 Determinant of the n X n matrix M_(i,j)=i/gcd(i,j)=lcm(i,j)/j.

1, -1, 2, -2, 8, 16, -96, 96, -192, -768, 7680, 15360, -184320, -1105920, -8847360, 8847360, -141557760, -283115520, 5096079360, 20384317440, 244611809280, 2446118092800, -53814598041600, -107629196083200, 430516784332800, 5166201411993600, -10332402823987200

Offset: 1

Author

Benoit Cloitre, Aug 19 2003

Keywords

• sign

Comments

Determinant of a symmetric polynomial evaluated at x = 1 that starts:

{{1, 1, 1, 1, 1},

{1, -1 + x, 1, -1 + x, 1},

{1, 1, -2 + x, 1, 1},

{1, -1 + x, 1, -1, 1},

{1, 1, 1, 1, -4 + x}}. - Mats Granvik, Jul 22 2012

Crossrefs

Cf. A000040, A000720, A023900.

Programs

Mathematica

Clear[nn, t, n, k, M, x]; nnn = 27; a = Range[nnn]*0; Do[nn = ii; t[n_, 1] = 1; t[1, k_] = 1; t[n_, k_] := t[n, k] = If[n < k, If[And[n > 1, k > 1], x - Sum[t[k - i, n], {i, 1, n - 1}], 0], If[And[n > 1, k > 1], x - Sum[t[n - i, k], {i, 1, k - 1}], 0]];  x = 1;  M = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];  a[[ii]] = Det[M], {ii, 1, nnn}]; a (* Mats Granvik, Jul 22 2012 *)

PARI

a(n)=matdet(matrix(n,n,i,j,i/gcd(i,j)))

Formula

a(n+1)/a(n) = A023900(n+1) the reciprocity balance of n+1.

a(n) = Product_{i=1..A000720(n)} (1-A000040(i))^floor(n/A000040(i)). - Enrique Pérez Herrero, Jul 12 2012