A099451 - OEIS

%I #20 Jul 24 2022 06:12:32

%S 1,5,17,45,96,155,119,-365,-2217,-7360,-18791,-38435,-57639,-28875,

%T 200992,1015075,3179711,7796715,15240559,20915840,3218033,-103746315,

%U -458355231,-1362884995,-3211177504,-5977952405,-7345234233,2382397955,51340513351,204512766400,579756435849

%N A Chebyshev transform of A099450 associated to the knot 7_7.

%C The denominator is a parameterization of the Alexander polynomial for the knot 7_7. The g.f. is the image of the g.f. of A099450 under the Chebyshev transform A(x)->(1/(1+x^2))A(x/(1+x^2)).

%H Dror Bar-Natan, <a href="http://katlas.org/wiki/Main_Page">The Rolfsen Knot Table</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,5,-1).

%F G.f.: (1+x^2)/(1-5x+9x^2-5x^3+x^4).

%F a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k * Sum_{j=0..n-2k} C(n-2k-j, j)*(-7)^j*5^(n-2k-2j).

%F a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*A099450(n-2k).

%F a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*A099450(k)/2.

%F a(n) = Sum_{k even, 0<=k<=n} A099452(n-k). [corrected by _Kevin Ryde_, Jul 24 2022]

%Y Cf. A099450, A099452.

%K easy,sign

%O 0,2

%A _Paul Barry_, Oct 16 2004