A084780 a(n) = sum of absolute-valued coefficients of (1+3*x-x^2)^n.

1, 5, 21, 77, 291, 1119, 3523, 15007, 50923, 182669, 701121, 2379129, 8909361, 32490021, 106309861, 423990203, 1456199483, 5089398187, 19942506259, 65753622619, 252337832801, 903751067081, 3026099773993, 11771846189609

Offset: 0

Author

Paul D. Hanna, Jun 14 2003

Keywords

• nonn

Comments

The expansion of (1 + a*x - b*x^2)^n is: (1 + a*x - b*x^2)^n = Sum_{k=0..2*n} f(n, k)*x^k, where f(n, k) = (n!/(2*n-k)!) * (-b)^((k-n)/2) * (a^2 + 4*b)^(n/2) * LegendreP(n, n-k, a/sqrt(a^2 + 4*b)). - G. C. Greubel, Jun 04 2023

Links

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

Crossrefs

Cf. A084775, A084776, A084777, A084778, A084779.

Programs

Magma

m:=40;
R:=PowerSeriesRing(Integers(), 2*(m+2));
f:= func< n,k | Coefficient(R!( (1+3*x-x^2)^n ), k) >;
[(&+[ Abs(f(n,k)): k in [0..2*n]]): n in [0..m]]; // G. C. Greubel, Jun 04 2023

Mathematica

Table[Total[Abs[CoefficientList[Expand[(1+3x-x^2)^n],x]]],{n,0,30}] (* Harvey P. Dale, Jan 04 2012 *)

PARI

for(n=0,40,S=sum(k=0,2*n,abs(polcoeff((1+3*x-x^2)^n,k,x))); print1(S","))

SageMath

def f(n,k):
    P. = PowerSeriesRing(QQ)
    return P( (1+3*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 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)*(13)^(n/2)*LegendreP(n, n-k, 3/sqrt(13)).. - G. C. Greubel, Jun 04 2023