A061914 Let H_n = n-th Hilbert matrix; sequence gives 1 / ( det(H_n) * denominator(permanent(H_n)) ).

1, 1, 1, 27, 567, 1, 1, 1, 7, 9, 5103, 1275989841, 992436543, 48629390607, 169706648853, 40257567, 63, 1, 7, 31, 1, 3969, 25865973, 117649, 117649, 16807, 49, 9, 81, 117369, 59049, 33480783, 930196594089, 4238886345135097131, 169560200598623521407

Offset: 1

Author

Asher Auel, May 20 2001

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Permanent.

Crossrefs

Cf. A005249.

Programs

Maple

with(linalg): seq(1/(denom(permanent(hilbert(n)))*det(hilbert(n))), n=1..16);

Mathematica

Permanent[m_List] := With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m.v), Times @@ v]]; f[n_] := Block[{i = Table[1/(i + j - 1), {i, n}, {j, n}]}, 1/(Det[i]Denominator[Permanent[i]])]; Table[ f[n], {n, 1, 18}] (* Robert G. Wilson v, Feb 06 2004 *)

PARI

permRWN(a)=n=matsize(a)[1]; if(n==1,return(a[1,1])); n1=n-1; sg=1; m=1; nc=0; in=vector(n); x=in; for(i=1,n,x[i]=a[i,n]-sum(j=1,n,a[i,j])/2); p=prod(i=1,n,x[i]); while(m,sg=-sg; j=1; if((nc%2)!=0,j++; while(in[j-1]==0,j++)); in[j]=1-in[j]; nc+=2*in[j]-1; m=nc!=in[n1]; z=2*in[j]-1; for(i=1,n,x[i]+=z*a[i,j]); p+=sg*prod(i=1,n,x[i])); return(2*(2*(n%2)-1)*p)
for(n=1,23,a=mathilbert(n); print1(1/(matdet(a)*denominator(permRWN(a)))", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 10 2007

PARI

for(n=1, 25, a=mathilbert(n); print1(1 / (matdet(a) * denominator(matpermanent(a)))", ")) \\ Vaclav Kotesovec, Aug 13 2021

Formula

a(n) = 1/(denominator(permanent(hilbert(n)))*det(hilbert(n))), where hilbert(n) denotes the n-th Hilbert matrix.

Extensions

a(18)-a(20) from Robert G. Wilson v, Feb 09 2004

a(21) from Eric W. Weisstein, Feb 19 2004

a(22) and a(23) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 10 2007

a(24)-a(34) from Vaclav Kotesovec, Aug 14 2021

a(35) from Vaclav Kotesovec, Aug 16 2021