A084778 a(n) = sum of absolute-valued coefficients of (1+2*x-3*x^2)^n.

1, 6, 28, 128, 660, 3016, 13108, 64112, 304068, 1332992, 6514356, 29341384, 131904528, 623547112, 2990903464, 13436119424, 61647598484, 284398511848, 1302463169256, 6195158123688, 28653898573420, 130138400720504

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, A084779, A084780.

Programs

Magma

m:=40;
R:=PowerSeriesRing(Integers(), 2*(m+2));
f:= func< n,k | Coefficient(R!( (1+2*x-3*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-3*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-3*x^2)^n,k,x))); print1(S","))

SageMath

def f(n,k):
    P. = PowerSeriesRing(QQ)
    return P( (1+2*x-3*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) = 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