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%I #10 Jan 31 2022 13:37:06
%S 0,0,0,1,4,15,36,84,160,300,500,825,1260,1911,2744,3920,5376,7344,
%T 9720,12825,16500,21175,26620,33396,41184,50700,61516,74529,89180,
%U 106575,126000,148800,174080,203456,235824,273105,313956,360639,411540,469300,532000,602700
%N a(n) = n*(1 - (-1)^n - 2*(3 + (-1)^n)*n^2 + 2*n^4)/384.
%C Definitions: (Start)
%C The k-th exterior power of a vector space V of dimension n is a vector subspace spanned by elements, called k-vectors, that are the exterior product of k vectors v_i in V.
%C Given a square matrix A that describes the vectors v_i in terms of a basis of V, the k-th exterior power of the matrix A is the matrix that represents the k-vectors in terms of the basis of V. (End)
%C Conjectures: (Start)
%C For n > 1, a(n) is the absolute value of the trace of the 3rd exterior power of an n X n square matrix M(n) defined as M[i,j] = floor((j - i + 1)/2). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-3)] in the characteristic polynomial of the matrix M(n), or the absolute value of the sum of all principal minors of M(n) of size 3.
%C For k > 3, the trace of the k-th exterior power of the matrix M(n) is equal to zero. (End)
%C The matrix M(n) is the n-th principal submatrix of the array A010751.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Characteristic_polynomial">Characteristic polynomial</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Exterior_algebra">Exterior algebra</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-8,-2,12,-2,-8,3,2,-1).
%F O.g.f.: x^3*(1 + 2*x + 4*x^2 + 2*x^3 + x^4)/((1 - x)^6*(1 + x)^4).
%F E.g.f.: x*(x*(x^3 + 10*x^2 + 23*x + 3)*cosh(x) + (x^4 + 10*x^3 + 21*x^2 + 9*x - 3)*sinh(x))/192.
%F a(n) = 2*a(n-1) + 3*a(n-2) - 8*a(n-3) - 2*a(n-4) + 12*a(n-5) - 2*a(n-6) - 8*a(n-7) + 3*a(n-8) + 2*a(n-9) - a(n-10) for n > 9.
%F a(2*n+1) = A108674(n-1) for n > 0.
%F Sum_{n>2} 1/a(n) = 192*log(2) - 6*zeta(3) - 249/2 = 1.371917248551933695710...
%t Table[n(1 - (-1)^n - 2*(3 + (-1)^n)n^2 + 2n^4)/384,{n,0,41}]
%Y Cf. A002620 (elements sum of the matrix M(n)), A010751, A108674, A350549 (permanent of the matrix M(n)).
%K nonn,easy
%O 0,5
%A _Stefano Spezia_, Jan 12 2022
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