cp's OEIS Frontend

A099454 A Chebyshev transform of A099453 associated to the knot 8_12.

%I A099454 #24 Jan 15 2025 04:18:43
%S A099454 1,7,37,175,792,3521,15539,68369,300431,1319472,5793745,25437727,
%T A099454 111681277,490315231,2152620360,9450575729,41490490763,182153978153,
%U A099454 799702876895,3510901281888,15413758929889,67670362004791,297090274041301,1304302623454159,5726223576745848
%N A099454 A Chebyshev transform of A099453 associated to the knot 8_12.
%C A099454 The denominator is a parameterization of the Alexander polynomial for the knot 8_12. The g.f. is the image of the g.f. of A099453 under the Chebyshev transform A(x)->(1/(1+x^2))A(x/(1+x^2)).
%H A099454 Stefano Spezia, <a href="/A099454/b099454.txt">Table of n, a(n) for n = 0..1500</a>
%H A099454 Dror Bar-Natan, <a href="http://katlas.org/wiki/Main_Page">The Rolfsen Knot Table</a>.
%H A099454 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-13,7,-1).
%F A099454 G.f.: (1+x^2)/(1-7x+13x^2-7x^3+x^4).
%F A099454 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)*(-11)^j*7^(n-2*k-2*j).
%F A099454 a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*A099453(n-2*k).
%F A099454 a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*A099453(k)/2.
%F A099454 a(n) = Sum_{k=0..n} A099455(n-k)*binomial(1, k/2)*(1+(-1)^k)/2.
%t A099454 CoefficientList[Series[(1+x^2)/(1-7x+13x^2-7x^3+x^4),{x,0,24}],x] (* _Stefano Spezia_, May 13 2024 *)
%Y A099454 Cf. A099453, A099455.
%K A099454 easy,nonn
%O A099454 0,2
%A A099454 _Paul Barry_, Oct 16 2004