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a(n)=(-1)^n(1 - (1/12)n(n + 1)(12 - n + n^2)).
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%I #5 Jun 16 2023 17:21:51

%S 1,1,-6,17,-39,79,-146,251,-407,629,-934,1341,-1871,2547,-3394,4439,

%T -5711,7241,-9062,11209,-13719,16631,-19986,23827,-28199,33149,-38726,

%U 44981,-51967,59739,-68354,77871,-88351,99857,-112454,126209,-141191,157471,-175122,194219,-214839,237061,-260966

%N a(n)=(-1)^n(1 - (1/12)n(n + 1)(12 - n + n^2)).

%C a(n) is the determinant of the n X n matrix M with M(i,i)=2i-1, M(i,j)=i+j.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-5,-10,-10,-5,-1).

%F G.f.: (1 + 6x + 9x^2 + 7x^3 + x^4)/(1 + x)^5.

%F a(n) = (-1)^n * (1 - A002378(n) - A002415(n)).

%t CoefficientList[Series[(1 + 6x + 9x^2 + 7x^3 + x^4)/(1 + x)^5, {x, 0, 50}], x]

%t LinearRecurrence[{-5,-10,-10,-5,-1},{1,1,-6,17,-39},50] (* _Harvey P. Dale_, Jun 16 2023 *)

%Y Cf. A002378, A002415.

%K easy,sign

%O 0,3

%A Mario Catalani (mario.catalani(AT)unito.it), Feb 12 2003