A122192 - OEIS

%I #12 Jul 12 2021 02:40:46

%S 6,0,-1518,-7728,1149126,9775920,-851127150,-10374206304,619950551814,

%T 10059106207584,-443172509029998,-9223980220220304,309985135145332422,

%U 8134978519171135632,-211181377213616588526,-6965969413257227260608,139095682365347347024902

%N Sum of the n-th powers of the roots of the (reduced) weight enumerator of the extended Golay code (1 + 759*x^2 + 2576*x^3 + 759*x^4 + x^6).

%H G. C. Greubel, <a href="/A122192/b122192.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,-759,-2576,-759,0,-1).

%F G.f.: 6*(1 + 506*x^2 + 1288*x^3 + 253*x^4)/(1 + 759*x^2 + 2576*x^3 + 759*x^4 + x^6).

%p Newt:=proc(f) local t1,t2,t3,t4; t1:=f; t2:=diff(f,x); t3:=expand(x^degree(t1,x)*subs(x=1/x,t1)); t4:=expand(x^degree(t2,x)*subs(x=1/x,t2)); factor(t4/t3); end;

%p g:=1+759*x^2+2576*x^3+759*x^4+x^6; Newt(g); series(%,x,60);

%t LinearRecurrence[{0,-759,-2576,-759,0,-1}, {6,0,-1518,-7728,1149126,9775920}, 30] (* _G. C. Greubel_, Jul 11 2021 *)

%o (PARI) polsym(x^6 + 759*x^4 + 2576*x^3 + 759*x^2 + 1, 30) \\ _Charles R Greathouse IV_, Jul 20 2016

%o (Sage)

%o def A122192_list(prec):

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

%o     return P( 6*(1+506*x^2+1288*x^3+253*x^4)/(1+759*x^2+2576*x^3+759*x^4 +x^6) ).list()

%o A122192_list(30) # _G. C. Greubel_, Jul 11 2021

%Y Cf. A001380, A123013.

%K sign,easy

%O 0,1

%A _N. J. A. Sloane_, Nov 12 2006