determinant

One problem with determinants

(%i1) h[i,j]:=if i=j then i
        elseif abs(i-j)=1 then 1
        else 0$
(%i2) A(n):=genmatrix(lambda([i,j], h[i,j]), n, n)$
(%i3) A(5);
(%o3) matrix([1,1,0,0,0],[1,2,1,0,0],[0,1,3,1,0],[0,0,1,4,1],[0,0,0,1,5])
(%i4) makelist(determinant(A(k)),k,1,10);
(%o4) [1,1,2,7,33,191,1304,10241,90865,898409]

 *****************

(%i5) f(n):= if n=1  or  n=2 then 1 else n*f(n-1)-f(n-2)$
(%i6) makelist(f(k),k,1,10);
(%o6) [1,1,2,7,33,191,1304,10241,90865,898409]

 ******************

(%i7) load(contrib_ode)$
(%i8) a(n):=(bessel_j(n+1,2)*bessel_y(0,2)-bessel_j(0,2)*bessel_y(n+1,2))/
      (bessel_j(1,2)*bessel_y(0,2)-bessel_j(0,2)*bessel_y(1,2))$
(%i9) makelist(a(k),k,1,10)$ besjsimp(%)$ besysimp(%);
(%o11) [1,1,2,7,33,191,1304,10241,90865,898409]


Prove   that    a(n) = f(n) = determinant(A(n)).

best regards

Aleksas D