A061914 - OEIS

%I #27 Nov 22 2024 19:16:54

%S 1,1,1,27,567,1,1,1,7,9,5103,1275989841,992436543,48629390607,

%T 169706648853,40257567,63,1,7,31,1,3969,25865973,117649,117649,16807,

%U 49,9,81,117369,59049,33480783,930196594089,4238886345135097131,169560200598623521407

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Permanent.html">Permanent.</a>

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

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

%t 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 *)

%o (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)

%o for(n=1,23,a=mathilbert(n); print1(1/(matdet(a)*denominator(permRWN(a)))", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 10 2007

%o (PARI) for(n=1, 25, a=mathilbert(n); print1(1 / (matdet(a) * denominator(matpermanent(a)))", ")) \\ _Vaclav Kotesovec_, Aug 13 2021

%Y Cf. A005249.

%K nonn

%O 1,4

%A _Asher Auel_, May 20 2001

%E a(18)-a(20) from _Robert G. Wilson v_, Feb 09 2004

%E a(21) from _Eric W. Weisstein_, Feb 19, 2004

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

%E a(24)-a(34) from _Vaclav Kotesovec_, Aug 14 2021

%E a(35) from _Vaclav Kotesovec_, Aug 16 2021