A350689 a(n) = n*(1 - (-1)^n - 2*(3 + (-1)^n)*n^2 + 2*n^4)/384.

0, 0, 0, 1, 4, 15, 36, 84, 160, 300, 500, 825, 1260, 1911, 2744, 3920, 5376, 7344, 9720, 12825, 16500, 21175, 26620, 33396, 41184, 50700, 61516, 74529, 89180, 106575, 126000, 148800, 174080, 203456, 235824, 273105, 313956, 360639, 411540, 469300, 532000, 602700

Offset: 0

Author

Stefano Spezia, Jan 12 2022

Keywords

• nonn
• easy

Comments

Definitions: (Start)

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.

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)

Conjectures: (Start)

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.

For k > 3, the trace of the k-th exterior power of the matrix M(n) is equal to zero. (End)

The matrix M(n) is the n-th principal submatrix of the array A010751.

Links

• Wikipedia, Characteristic polynomial
• Wikipedia, Exterior algebra
• Index entries for linear recurrences with constant coefficients, signature (2,3,-8,-2,12,-2,-8,3,2,-1).

Crossrefs

Cf. A002620 (elements sum of the matrix M(n)), A010751, A108674, A350549 (permanent of the matrix M(n)).

Programs

Mathematica

Table[n(1 - (-1)^n - 2*(3 + (-1)^n)n^2 + 2n^4)/384,{n,0,41}]

Formula

O.g.f.: x^3*(1 + 2*x + 4*x^2 + 2*x^3 + x^4)/((1 - x)^6*(1 + x)^4).

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.

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.

a(2*n+1) = A108674(n-1) for n > 0.

Sum_{n>2} 1/a(n) = 192*log(2) - 6*zeta(3) - 249/2 = 1.371917248551933695710...