A211158 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and positive odd determinant.

20, 84, 528, 1040, 3060, 4788, 10304, 14400, 26100, 34100, 55440, 69264, 104468, 126420, 180480, 213248, 291924, 338580, 448400, 512400, 660660, 745844, 940608, 1051200, 1301300, 1441908, 1756944, 1932560, 2322900, 2538900, 3015680, 3277824, 3852948, 4167380

Offset: 1

Author

Clark Kimberling, Apr 05 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A210000.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1, 4, -4, -6, 6, 4, -4, -1, 1).

Crossrefs

Cf. A210000.

Programs

Magma

[n*(n+1)*(3*n+1+3*n^2-(-1)^n*(2*n+1)): n in [1..35]]; // Vincenzo Librandi, Dec 14 2016

Mathematica

a = -n; b = n; z1 = 25;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
u[n_] := u[n] = Sum[c[n, 2 k], {k, 0, 2*n^2}]
v[n_] := v[n] = Sum[c[n, 2 k], {k, 1, 2*n^2}]
w[n_] := w[n] = Sum[c[n, 2 k - 1], {k, 1, 2*n^2}]
u1 = Table[u[n], {n, 1, z1}] (* A211156 *)
v1 = Table[v[n], {n, 1, z1}] (* A211157 *)
w1 = Table[w[n], {n, 1, z1}] (* A211158 *)
(u1 - 1)/4 (* integers *)
v1/4 (* integers *)
w1/4 (* integers *)
Table[n*(n+1)*(3*n+1+3*n^2-(-1)^n*(2*n+1)),{n,35}] (* Vincenzo Librandi, Dec 14 2016 *)
CoefficientList[ Series[-(( 4(5 + 16x + 91x^2 + 64x^3 + 91x^4 + 16x^5 + 5x^6))/((x -1)^5 (x +1)^4)), {x, 0, 35}], x] (* or *)
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {20, 84, 528, 1040, 3060, 4788, 10304, 14400, 26100}, 36] (* Robert G. Wilson v, Dec 14 2016 *)

Python

def A211158(n):
    return n*(n+1)*(3*n+1+3*n**2-(-1)**n*(2*n+1)) # Chai Wah Wu, Dec 13 2016

Formula

From Chai Wah Wu, Dec 13 2016: (Start)

For n >= 0:

a(n) = A211155(n)/2.

a(n) = n*(n + 1)*(3*n + 1 + 3*n^2 - (-1)^n*(2*n + 1)). Therefore:

a(n) = n^2*(n + 1)*(3*n + 1) if n is even,

a(n) = n*(n + 1)^2*(3*n + 2) if n is odd.

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.

G.f.: x*(-20*x^6 - 64*x^5 - 364*x^4 - 256*x^3 - 364*x^2 - 64*x - 20)/((x - 1)^5*(x + 1)^4). (End)

a(n) = a(-n-1). - Bruno Berselli, Dec 14 2016