A084611 a(n) = sum of absolute values of coefficients of (1+x-x^2)^n.

1, 3, 7, 13, 35, 83, 165, 367, 899, 1957, 3839, 9771, 22709, 43213, 102963, 255061, 525601, 1098339, 2798273, 6202969, 11746259, 29976073, 70898649, 140495779, 314391789, 787757461, 1688887719, 3337986541, 8583687613, 19647782463

Offset: 0

Author

Paul D. Hanna, Jun 01 2003

Keywords

• nonn

Comments

Limit_{n -> oo} a(n+1)/a(n) does not exist; however, lim_{n -> oo} a(n)^(1/n) = sqrt(5) (conjecture).

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500
• Vaclav Kotesovec, Asymptotic of sequence A084611, Jul 26 2013.

Crossrefs

Cf. A002426, A084600 - A084610, A084612 - A084615.

Programs

Magma

A084610:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-1)^j: j in [0..k]]) >;
[(&+[Abs(A084610(n,k)): k in [0..2*n]]): n in [0..50]]; // G. C. Greubel, Mar 26 2023

Mathematica

Table[Sum[Abs[Coefficient[Expand[(1+x-x^2)^n],x,k]],{k,0,2*n}],{n,0,30}] (* Vaclav Kotesovec, Jul 28 2013 *)

PARI

{a(n)=sum(k=0,2*n,abs(polcoeff((1+x-x^2+x*O(x^k))^n,k)))}
for(n=0,30,print1(a(n),", "))

SageMath

def A084610(n,k): return sum(binomial(n,j)*binomial(n-j,k-2*j)*(-1)^j for j in range(k//2+1))
def A084611(n): return 2*sum(abs(A084610(n,k)) for k in range(n)) + abs(A084610(n,n))
[A084611(n) for n in range(50)] # G. C. Greubel, Mar 26 2023