|
| |
|
|
A087168
|
|
Expansion of (1+2*x)/(1+3*x+4*x^2).
|
|
4
| |
|
|
1, -1, -1, 7, -17, 23, -1, -89, 271, -457, 287, 967, -4049, 8279, -8641, -7193, 56143, -139657, 194399, -24569, -703889, 2209943, -3814273, 2603047, 7447951, -32756041, 68476319, -74404793, -50690897, 449691863, -1146312001, 1640168551, -335257649, -5554901257
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| For positive n, a(n) equals 2^n times the permanent of the (2n)X(2n) tridiagonal matrix with 1/sqrt(2)'s along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). [From John M. Campbell, Jul 08 2011]
For n>3, equals -1 times the determinant of the (n-2)X(n-2) matrix with 2^2's along the superdiagonal, 3^2's along the main diagonal, 4^2's along the subdiagonal, etc., and 0's everywhere else. [From John M. Campbell, Dec 01 2011]
|
|
|
FORMULA
| G.f.: (1+2*x)/(1+3*x+4*x^2).
a(n) = -3*a(n-1)-4*a(n-2); a(0)=1, a(1)=-1.
a(n) = sum(k=0..n, C(n+k,2*k)*(-2)^(n-k) ).
a(n) = (1/14)*I*sqrt(7)*[ -3/2-(1/2)*I*sqrt(7)]^n-(1/14)*I*sqrt(7)*[ -3/2+(1/2)*I *sqrt(7)]^n+(1/2)*[ -3/2+(1/2)*I*sqrt(7)]^n+(1/2)*[ -3/2-(1/2)*I*sqrt(7)]^n, with n>=0 and I=sqrt(-1). - Paolo P. Lava, Jun 12 2008
|
|
|
MATHEMATICA
| CoefficientList[Series[(1 + 2x)/(4x^2 + 3x + 1), {x, 0, 30}], x]
Table[-Det[Array[Sum[KroneckerDelta[#1, #2+q]*(q+3)^2, {q, -1, n-2}] &, {n-2, n-2}]], {n, 4, 50}] (* From John M. Campbell, Dec 01 2011 *)
|
|
|
PROG
| (MAGMA) a087168:=func< n | &+[ Binomial(n+k, 2*k)*(-2)^(n-k): k in [0..n] ] >; [ a087168(n): n in [0..35] ];
|
|
|
CROSSREFS
| Sequence in context: A144695 A125244 A070416 * A191087 A032454 A107643
Adjacent sequences: A087165 A087166 A087167 * A087169 A087170 A087171
|
|
|
KEYWORD
| easy,sign
|
|
|
AUTHOR
| Mario Catalani (mario.catalani(AT)unito.it), Aug 22 2003
|
| |
|
|
