A099460 - OEIS

%I #19 Sep 08 2022 08:45:15

%S 1,7,39,203,1040,5313,27133,138565,707643,3613904,18456077,94254531,

%T 481354555,2458260679,12554250288,64114111901,327428500337,

%U 1672165762785,8539691368807,43611901581472,222724437852585

%N A Chebyshev transform of A099459 associated to the knot 9_48.

%C The denominator is a parameterization of the Alexander polynomial for the knot 9_48. The g.f. is the image of the g.f. of A099459 under the Chebyshev transform A(x) -> (1/(1+x^2))*A(x/(1+x^2)).

%H G. C. Greubel, <a href="/A099460/b099460.txt">Table of n, a(n) for n = 0..1000</a>

%H Dror Bar-Natan, <a href="http://katlas.org/wiki/9_48">9 48</a>, The Knot Atlas.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-11,7,-1).

%F G.f.: (1+x^2)/(1 -7*x +11*x^2 -7*x^3 +x^4).

%F a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*( Sum_{j=0..n-2*k} C(n-2*k-j, j)(-9)^j*7^(n-2*k-2*j) ).

%F a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)(-1)^k*A099459(n-2*k).

%F a(n) = (1/2)*Sum_{k=0..n} (-1)^((n-k)/2)*(1 + (-1)^(n+k))*binomial((n+k)/2, k) *A099459(k).

%F a(n) = Sum_{k=0..n} A099461(n-k)*binomial(1, k/2)*((1+(-1)^k)/2).

%t LinearRecurrence[{7,-11,7,-1}, {1,7,39,203}, 30] (* _G. C. Greubel_, Nov 18 2021 *)

%o (Magma) I:=[1,7,39,203]; [n le 4 select I[n] else 7*Self(n-1) - 11*Self(n-2) +7*Self(n-3) -Self(n-4): n in [1..31]]; // _G. C. Greubel_, Nov 18 2021

%o (Sage)

%o def A099460_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (1+x^2)/(1-7*x+11*x^2-7*x^3+x^4) ).list()

%o A099460_list(30) # _G. C. Greubel_, Nov 18 2021

%Y Cf. A099459, A099461.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 16 2004