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A330700 a(n) = (n - 1)*n*(2*n^2 + 4*n - 1)/6.

%I A330700 #22 Sep 08 2022 08:46:24
%S A330700 0,0,5,29,94,230,475,875,1484,2364,3585,5225,7370,10114,13559,17815,
%T A330700 23000,29240,36669,45429,55670,67550,81235,96899,114724,134900,157625,
%U A330700 183105,211554,243194,278255,316975,359600,406384,457589,513485,574350,640470,712139,789659
%N A330700 a(n) = (n - 1)*n*(2*n^2 + 4*n - 1)/6.
%C A330700 Conjectures: (Start)
%C A330700 For n > 1, a(n) is the absolute value of the trace of the 2nd exterior power of an n X n square matrix M(n) defined as M[i,j,n] = i*j if i < 3 or j < 3 and M[i,j,n] = 2*(i + j) - 4 otherwise (see A330287). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of M(n), or the absolute value of the sum of all principal minors of M(n) of size 2.
%C A330700 For k > 2, the trace of the k-th exterior power of M(n) is equal to zero.
%C A330700 (End)
%H A330700 Wikipedia, <a href="https://en.wikipedia.org/wiki/Characteristic_polynomial">Characteristic polynomial</a>
%H A330700 Wikipedia, <a href="https://en.wikipedia.org/wiki/Exterior_algebra">Exterior algebra</a>
%H A330700 Wikipedia, <a href="https://en.wikipedia.org/wiki/Harmonic_number">Harmonic number</a>
%H A330700 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A330700 O.g.f.: x^2*(5 + 4*x - x^2)/(1 - x)^5.
%F A330700 E.g.f.: exp(x)*x^2*(15 + 14*x + 2*x^2)/6.
%F A330700 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
%F A330700 a(n) = A000217(n-1)*A268581(n-1)/3 for n > 0.
%F A330700 Sum_{k>=2} 1/a(k) = (1/5)*((18 + 7*sqrt(6))*H(2-sqrt(3/2)) + (18 - 7*sqrt(6))*H(2+sqrt(3/2)) - 30) = 0.254905801002729039998040617... where H(x) = Integral_{t=0..1} (1 - t^x)/(1 - t) dt is the function that interpolates the harmonic numbers.
%t A330700 Table[(n-1)n(2n^2+4n-1)/6,{n,0,39}]
%o A330700 (Magma) I:=[0, 0, 5, 29, 94]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]];
%o A330700 (PARI) my(x='x + O('x^39)); concat([0, 0], Vec(serlaplace((1/6)*exp(x)*x^2*(15+14*x+2*x^2))))
%o A330700 (Sage) (x^2*(5+4*x-x^2)/(1-x)^5).series(x, 40).coefficients(x, sparse=False)
%Y A330700 Cf. A000217, A001105 (super- and subdiagonal sum of M(n)), A001844 (trace of M(n)), A005843 (antitrace of M(n)), A268581, A319840, A322844, A330287 (permanent of M(n)).
%K A330700 easy,nonn
%O A330700 0,3
%A A330700 _Stefano Spezia_, Dec 26 2019