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(-1)^(n+1) * Determinant of n X n matrix of form [1^2 2^2 3^2 4^2 5^2 / 2^2 1^2 2^2 3^2 4^2 / 3^2 2^2 1^2 2^2 3^2 / 4^2 3^2 2^2 1^2 2^2 / 5^2 4^2 3^2 2^2 1^2].
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%I #12 Aug 13 2024 15:39:06

%S 1,15,176,1680,13824,102144,696320,4460544,27197440,159318016,

%T 902823936,4975493120,26776436736,141180272640,731218182144,

%U 3728300048384,18747532247040,93110596009984,457328117678080,2223830986653696,10715840324304896,51209754063667200

%N (-1)^(n+1) * Determinant of n X n matrix of form [1^2 2^2 3^2 4^2 5^2 / 2^2 1^2 2^2 3^2 4^2 / 3^2 2^2 1^2 2^2 3^2 / 4^2 3^2 2^2 1^2 2^2 / 5^2 4^2 3^2 2^2 1^2].

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (20,-160,640,-1280,1024).

%F a(n) = 2^(-5+2*n)*(6+8*n+5*n^2+4*n^3+n^4)/3. - _Colin Barker_, Oct 24 2014

%F G.f.: -x*(64*x^4-80*x^3+36*x^2-5*x+1) / (4*x-1)^5. - _Colin Barker_, Oct 24 2014

%t LinearRecurrence[{20,-160,640,-1280,1024},{1,15,176,1680,13824},30] (* _Harvey P. Dale_, Aug 13 2024 *)

%o (PARI) for(n=1,30,print1((-1)^(n+1)*matdet(matrix(n,n,i,j,sum(k=0,n-1,if(abs(i-j)-k,0,(k+1)^2)))),","))

%o (PARI) Vec(-x*(64*x^4-80*x^3+36*x^2-5*x+1)/(4*x-1)^5 + O(x^100)) \\ _Colin Barker_, Oct 24 2014

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_, Jun 20 2002

%E More terms, and typos in data fixed by _Colin Barker_, Oct 24 2014