%I #14 Jun 04 2023 03:55:55

%S 1,4,12,36,100,300,776,2412,6304,19036,50952,148896,393452,1211444,

%T 3167004,9672772,25295248,76084796,200590424,608621376,1617201648,

%U 4908511140,12658776540,38907904188,102775961200,310485090044

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

%H G. C. Greubel, <a href="/A084776/b084776.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(2) - 1)^k * Sum_{j=0..k} binomial(n, j)*binomial(n, k-j)*(-1)^j*(1+sqrt(2))^(2*j). - _G. C. Greubel_, Jun 03 2023

%t T[n_, k_]:=T[n,k]=SeriesCoefficient[Series[(1+2*x-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 03 2023 *)

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

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

%o (SageMath)

%o def f(n,k):

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

%o return P( (1+2*x-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. A084775, A084777, A084778, A084779, A084780.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 14 2003