A084611 - OEIS

%I #21 Apr 07 2023 17:08:20

%S 1,3,7,13,35,83,165,367,899,1957,3839,9771,22709,43213,102963,255061,

%T 525601,1098339,2798273,6202969,11746259,29976073,70898649,140495779,

%U 314391789,787757461,1688887719,3337986541,8583687613,19647782463

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

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

%H Paul D. Hanna, <a href="/A084611/b084611.txt">Table of n, a(n) for n = 0..500</a>

%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of sequence A084611</a>, Jul 26 2013.

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

%o (PARI) {a(n)=sum(k=0,2*n,abs(polcoeff((1+x-x^2+x*O(x^k))^n,k)))}

%o for(n=0,30,print1(a(n),", "))

%o (Magma)

%o A084610:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-1)^j: j in [0..k]]) >;

%o [(&+[Abs(A084610(n,k)): k in [0..2*n]]): n in [0..50]]; // _G. C. Greubel_, Mar 26 2023

%o (SageMath)

%o def A084610(n,k): return sum(binomial(n,j)*binomial(n-j,k-2*j)*(-1)^j for j in range(k//2+1))

%o def A084611(n): return 2*sum(abs(A084610(n,k)) for k in range(n)) + abs(A084610(n,n))

%o [A084611(n) for n in range(50)] # _G. C. Greubel_, Mar 26 2023

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 01 2003