%I #16 Feb 16 2021 17:54:40

%S 1,-1,2,-2,8,16,-96,96,-192,-768,7680,15360,-184320,-1105920,-8847360,

%T 8847360,-141557760,-283115520,5096079360,20384317440,244611809280,

%U 2446118092800,-53814598041600,-107629196083200,430516784332800,5166201411993600,-10332402823987200

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

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

%C {{1, 1, 1, 1, 1},

%C {1, -1 + x, 1, -1 + x, 1},

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

%C {1, -1 + x, 1, -1, 1},

%C {1, 1, 1, 1, -4 + x}}. - _Mats Granvik_, Jul 22 2012

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

%F a(n) = Product_{i=1..A000720(n)} (1-A000040(i))^floor(n/A000040(i)). - _Enrique Pérez Herrero_, Jul 12 2012

%t 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 *)

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

%Y Cf. A000040, A000720, A023900.

%K sign

%O 1,3

%A _Benoit Cloitre_, Aug 19 2003