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

1,1

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).

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

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 *)

Prog

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

Crossrefs

Cf. A210000.

Sequence in context: A006566 A205312 A268888 * A154077 A027849 A243645

Adjacent sequences: A211155 A211156 A211157 * A211159 A211160 A211161

Keyword

nonn,easy

Author

Clark Kimberling, Apr 05 2012

Status

approved