cp's OEIS Frontend

A084779 a(n) = sum of absolute-valued coefficients of (1+2*x-4*x^2)^n.

%I A084779 #9 Jun 05 2023 08:52:41
%S A084779 1,7,41,207,1313,7807,42593,232463,1290433,7604415,42034721,236031231,
%T A084779 1363681121,7457831007,39670144513,231087069823,1291433872385,
%U A084779 7373001299199,41437235793921,229538650588863,1268719471103233
%N A084779 a(n) = sum of absolute-valued coefficients of (1+2*x-4*x^2)^n.
%H A084779 G. C. Greubel, <a href="/A084779/b084779.txt">Table of n, a(n) for n = 0..1000</a>
%F A084779 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 A084779 T[n_,k_]:=T[n,k]=SeriesCoefficient[Series[(1+2*x-4*x^2)^n,{x,0,2n}],k];
%t A084779 a[n_]:= a[n]= Sum[Abs[T[n,k]], {k,0,2n}];
%t A084779 Table[a[n], {n,0,40}] (* _G. C. Greubel_, Jun 04 2023 *)
%o A084779 (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 A084779 (Magma)
%o A084779 m:=40;
%o A084779 R<x>:=PowerSeriesRing(Integers(), 2*(m+2));
%o A084779 f:= func< n,k | Coefficient(R!( (1+2*x-4*x^2)^n ), k) >;
%o A084779 [(&+[ Abs(f(n,k)): k in [0..2*n]]): n in [0..m]]; // _G. C. Greubel_, Jun 04 2023
%o A084779 (SageMath)
%o A084779 def f(n,k):
%o A084779     P.<x> = PowerSeriesRing(QQ)
%o A084779     return P( (1+2*x-4*x^2)^n ).list()[k]
%o A084779 def a(n): return sum( abs(f(n,k)) for k in range(2*n+1) )
%o A084779 [a(n) for n in range(41)] # _G. C. Greubel_, Jun 04 2023
%Y A084779 Cf. A084775, A084776, A084777, A084778, A084780.
%K A084779 nonn
%O A084779 0,2
%A A084779 _Paul D. Hanna_, Jun 14 2003