|
%I #9 Jun 04 2023 02:15:21
%S 1,6,28,128,660,3016,13108,64112,304068,1332992,6514356,29341384,
%T 131904528,623547112,2990903464,13436119424,61647598484,284398511848,
%U 1302463169256,6195158123688,28653898573420,130138400720504
%N a(n) = sum of absolute-valued coefficients of (1+2*x-3*x^2)^n.
%H G. C. Greubel, <a href="/A084778/b084778.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) = Sum_{j=0..k} binomial(n, j)*binomial(n, k-j)*(-3)^j = binomial(n, k)*Hypergeometric2F1([-n, -k], [n-k+1], -3) = (n!/k!)*4^n*(-3)^((k-n)/2)*LegendreP(n, k-n, -1/2). - _G. C. Greubel_, Jun 04 2023
%t T[n_,k_]:=T[n,k]=SeriesCoefficient[Series[(1+2*x-3*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-3*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-3*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-3*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, A084779, A084780.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 14 2003
|
