cp's OEIS Frontend

A084775 a(n) = sum of absolute valued coefficients of (1+x-4*x^2)^n.

%I A084775 #15 Jun 04 2023 02:22:05
%S A084775 1,6,34,184,956,4776,22986,118304,624634,3281346,17687330,92606914,
%T A084775 470392898,2348031430,11932314170,62345998488,326780375778,
%U A084775 1691296908076,8780141027670,45168987187058,230213109996786
%N A084775 a(n) = sum of absolute valued coefficients of (1+x-4*x^2)^n.
%H A084775 G. C. Greubel, <a href="/A084775/b084775.txt">Table of n, a(n) for n = 0..1000</a>
%F A084775 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 A084775 T[n_, k_]:=T[n,k]=SeriesCoefficient[Series[(1+x-2*x^2)^n,{x,0,2n}], k];
%t A084775 a[n_]:= a[n]= Sum[Abs[T[[k+1]]], {k,0,2n}];
%t A084775 Table[a[n], {n,0,40}] (* _G. C. Greubel_, Jun 03 2023 *)
%o A084775 (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 A084775 (Magma)
%o A084775 R<x>:=PowerSeriesRing(Integers(), 100);
%o A084775 f:= func< n,k | Coefficient(R!( (1+x-4*x^2)^n ), k) >;
%o A084775 [(&+[ Abs(f(n,k)): k in [0..2*n]]): n in [0..40]]; // _G. C. Greubel_, Jun 03 2023
%o A084775 (SageMath)
%o A084775 def f(n,k):
%o A084775     P.<x> = PowerSeriesRing(QQ)
%o A084775     return P( (1+x-4*x^2)^n ).list()[k]
%o A084775 def a(n): return sum( abs(f(n,k)) for k in range(2*n+1) )
%o A084775 [a(n) for n in range(41)] # _G. C. Greubel_, Jun 03 2023
%Y A084775 Cf. A084776, A084777, A084778, A084779, A084780.
%K A084775 nonn
%O A084775 0,2
%A A084775 _Paul D. Hanna_, Jun 14 2003