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

0, 0, 5, 29, 94, 230, 475, 875, 1484, 2364, 3585, 5225, 7370, 10114, 13559, 17815, 23000, 29240, 36669, 45429, 55670, 67550, 81235, 96899, 114724, 134900, 157625, 183105, 211554, 243194, 278255, 316975, 359600, 406384, 457589, 513485, 574350, 640470, 712139, 789659

Offset: 0

Author

Stefano Spezia, Dec 26 2019

Keywords

• easy
• nonn

Comments

Conjectures: (Start)

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.

For k > 2, the trace of the k-th exterior power of M(n) is equal to zero.

(End)

Links

• Wikipedia, Characteristic polynomial
• Wikipedia, Exterior algebra
• Wikipedia, Harmonic number
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

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)).

Programs

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]];

Mathematica

Table[(n-1)n(2n^2+4n-1)/6,{n,0,39}]

PARI

my(x='x + O('x^39)); concat([0, 0], Vec(serlaplace((1/6)*exp(x)*x^2*(15+14*x+2*x^2))))

Sage

(x^2*(5+4*x-x^2)/(1-x)^5).series(x, 40).coefficients(x, sparse=False)

Formula

O.g.f.: x^2*(5 + 4*x - x^2)/(1 - x)^5.

E.g.f.: exp(x)*x^2*(15 + 14*x + 2*x^2)/6.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.

a(n) = A000217(n-1)*A268581(n-1)/3 for n > 0.

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.