A084776 a(n) = sum of absolute-valued coefficients of (1+2*x-x^2)^n.

1, 4, 12, 36, 100, 300, 776, 2412, 6304, 19036, 50952, 148896, 393452, 1211444, 3167004, 9672772, 25295248, 76084796, 200590424, 608621376, 1617201648, 4908511140, 12658776540, 38907904188, 102775961200, 310485090044

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, A084777, A084778, A084779, A084780.

Programs

Magma

m:=40;
R:=PowerSeriesRing(Integers(), 2*(m+2));
f:= func< n,k | Coefficient(R!( (1+2*x-x^2)^n ), k) >;
[(&+[ Abs(f(n,k)): k in [0..2*n]]): n in [0..m]]; // G. C. Greubel, Jun 03 2023

Mathematica

T[n_, k_]:=T[n,k]=SeriesCoefficient[Series[(1+2*x-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 03 2023 *)

PARI

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

SageMath

def f(n,k):
    P. = PowerSeriesRing(QQ)
    return P( (1+2*x-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 03 2023

Formula

a(n) = Sum_{k=0..2*n} abs(f(n, k)), where f(n, k) = (sqrt(2) - 1)^k * Sum_{j=0..k} binomial(n, j)*binomial(n, k-j)*(-1)^j*(1+sqrt(2))^(2*j). - G. C. Greubel, Jun 03 2023