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

1, 7, 41, 207, 1313, 7807, 42593, 232463, 1290433, 7604415, 42034721, 236031231, 1363681121, 7457831007, 39670144513, 231087069823, 1291433872385, 7373001299199, 41437235793921, 229538650588863, 1268719471103233

Offset: 0

Author

Paul D. Hanna, Jun 14 2003

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A084775, A084776, A084777, A084778, A084780.

Programs

Magma

m:=40;
R:=PowerSeriesRing(Integers(), 2*(m+2));
f:= func< n,k | Coefficient(R!( (1+2*x-4*x^2)^n ), k) >;
[(&+[ Abs(f(n,k)): k in [0..2*n]]): n in [0..m]]; // G. C. Greubel, Jun 04 2023

Mathematica

T[n_,k_]:=T[n,k]=SeriesCoefficient[Series[(1+2*x-4*x^2)^n,{x,0,2n}],k];
a[n_]:= a[n]= Sum[Abs[T[n,k]], {k,0,2n}];
Table[a[n], {n,0,40}] (* G. C. Greubel, Jun 04 2023 *)

PARI

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

SageMath

def f(n,k):
    P. = PowerSeriesRing(QQ)
    return P( (1+2*x-4*x^2)^n ).list()[k]
def a(n): return sum( abs(f(n,k)) for k in range(2*n+1) )
[a(n) for n in range(41)] # G. C. Greubel, Jun 04 2023

Formula

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