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A322844
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a(n) = (1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*(n mod 2)).
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3
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0, 0, 5, 6, 68, 50, 333, 196, 1040, 540, 2525, 1210, 5220, 2366, 9653, 4200, 16448, 6936, 26325, 10830, 40100, 16170, 58685, 23276, 83088, 32500, 114413, 44226, 153860, 58870, 202725, 76880, 262400, 98736, 334373, 124950, 420228, 156066, 521645, 192660, 640400, 235340
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OFFSET
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0,3
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COMMENTS
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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] = j + n*(i-1) if i is odd and M[i,j,n] = n*i - j + 1 if i is even (see A317614). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] 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 2.
For k > 2, the trace of the k-th exterior power of the matrix M(n) is equal to zero.
(End)
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LINKS
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FORMULA
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O.g.f.: -x^2*(5 + 6*x + 43*x^2 + 20*x^3 + 43*x^4 + 6*x^5 + 5*x^6)/((-1 + x)^5*(1 + x)^5)).
E.g.f.: (1/(12*x^2))*exp(-x)*(24 - 60*exp(x) + 21*x + 9*x^2 + 2*x^3 + x^4 + exp(2*x)*(36 - 33*x + 15*x^2 - 4*x^3 + 2*x^4)).
a(n) = (1/4)*n^2*(1 + n^2) for n even.
a(n) = (1/2)*A037270(n) for n even.
a(n) = (1/12)*(-1 + n)*n^2*(1 + n) for n odd.
a(2*n+1) = 5*a(2*n-1) - 10*a(2*n-3) + 10*a(2*n-5) - 5*a(2*n-7) + a(2*n-9), for n > 4.
a(2*n) = 5*a(2*n-2) - 10*a(2*n-4) + 10*a(2*n-6) - 5*a(2*n-8) + a(2*n-10), for n > 4.
O.g.f. for a(2*n+1): -x*(2*(3 + 10*x + 3*x^2))/(-1 + x)^5.
O.g.f. for a(2*n): x*(-5 - 43*x - 43*x^2 - 5*x^3)/(-1 + x)^5.
E.g.f. for a(2*n+1): (1/12)*(6*x*cosh(sqrt(x)) + sqrt(x)*(6 + x)*sinh(sqrt(x))).
E.g.f. for a(2*n): (1/4)*(x*(8 + x)*cosh(sqrt(x)) + 2*sqrt(x)*(1 + 3*x)*sinh(sqrt(x))).
Sum_{k>=1} 1/a(2*k) = (1/6)*(12 + Pi^2 - 6*Pi*coth(Pi/2)) = 0.21955691692893092525407699347398665248691900...
Sum_{k>=1} 1/a(2*k+1) = 3*(5 - Pi^2/2) = 0.1955933983659620717482635001857732970...
Sum_{k>=2} 1/a(k) = 17 - (4*Pi^2)/3 - Pi*coth(Pi/2) = 0.415150315294892997002340493659759949516369894...
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MAPLE
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a:=n->(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*modp(n, 2)): seq(a(n), n=0..50);
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MATHEMATICA
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a[n_]:=(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*Mod[n, 2]); Array[a, 50, 0]
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PROG
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(GAP) Flat(List([0..50], n->(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*(n mod 2))));
(Magma) [IsEven(n) select (1/4)*n^2*(1 + n^2) else (1/12)*(- 1 + n)*n^2*(1 + n): n in [0..50]];
(Maxima) a(n):=(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*mod(n, 2))$ makelist(a(n), n, 0, 50);
(PARI) a(n) = (1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*(n % 2));
(PARI) a(n) = abs(polcoeff(charpoly(matrix(n, n, i, j, if (i %2, j + n*(i-1), n*i - j + 1))), n-2)); \\ Michel Marcus, Feb 06 2019
(Python) [int(n**2*(3*(1 + n**2) - 2*(2 + n**2)*pow(n, 1, 2))/12) for n in range(0, 50)]
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CROSSREFS
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Cf. A317614 (trace of matrix M(n)).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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