%I #9 Jun 05 2023 08:52:41

%S 1,7,41,207,1313,7807,42593,232463,1290433,7604415,42034721,236031231,

%T 1363681121,7457831007,39670144513,231087069823,1291433872385,

%U 7373001299199,41437235793921,229538650588863,1268719471103233

%N a(n) = sum of absolute-valued coefficients of (1+2*x-4*x^2)^n.

%H G. C. Greubel, <a href="/A084779/b084779.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..2*n} abs(f(n, k)), where f(n, k) = (n!/(2*n-k)!)*i*(k-n)*2^k*5^(n/2)*LegendreP(n, n-k, 1/sqrt(5)). - _G. C. Greubel_, Jun 04 2023

%t T[n_,k_]:=T[n,k]=SeriesCoefficient[Series[(1+2*x-4*x^2)^n,{x,0,2n}],k];

%t a[n_]:= a[n]= Sum[Abs[T[n,k]], {k,0,2n}];

%t Table[a[n], {n,0,40}] (* _G. C. Greubel_, Jun 04 2023 *)

%o (PARI) for(n=0,40,S=sum(k=0,2*n,abs(polcoeff((1+2*x-4*x^2)^n,k,x))); print1(S","))

%o (Magma)

%o m:=40;

%o R<x>:=PowerSeriesRing(Integers(), 2*(m+2));

%o f:= func< n,k | Coefficient(R!( (1+2*x-4*x^2)^n ), k) >;

%o [(&+[ Abs(f(n,k)): k in [0..2*n]]): n in [0..m]]; // _G. C. Greubel_, Jun 04 2023

%o (SageMath)

%o def f(n,k):

%o P.<x> = PowerSeriesRing(QQ)

%o return P( (1+2*x-4*x^2)^n ).list()[k]

%o def a(n): return sum( abs(f(n,k)) for k in range(2*n+1) )

%o [a(n) for n in range(41)] # _G. C. Greubel_, Jun 04 2023

%Y Cf. A084775, A084776, A084777, A084778, A084780.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 14 2003