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a(n) = sum of absolute valued coefficients of (1+x-4*x^2)^n.
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%I #15 Jun 04 2023 02:22:05

%S 1,6,34,184,956,4776,22986,118304,624634,3281346,17687330,92606914,

%T 470392898,2348031430,11932314170,62345998488,326780375778,

%U 1691296908076,8780141027670,45168987187058,230213109996786

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

%H G. C. Greubel, <a href="/A084775/b084775.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) = ((sqrt(17) -1)/2)^k * Sum_{j=0..k} binomial(n, j)*binomial(n, k-j)*(-1)^j*((1+sqrt(17))/4 )^(2*j). - _G. C. Greubel_, Jun 03 2023

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

%t a[n_]:= a[n]= Sum[Abs[T[[k+1]]], {k,0,2n}];

%t Table[a[n], {n,0,40}] (* _G. C. Greubel_, Jun 03 2023 *)

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

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 100);

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

%o [(&+[ Abs(f(n,k)): k in [0..2*n]]): n in [0..40]]; // _G. C. Greubel_, Jun 03 2023

%o (SageMath)

%o def f(n,k):

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

%o return P( (1+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 03 2023

%Y Cf. A084776, A084777, A084778, A084779, A084780.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 14 2003