A084612 -id:A084612 - cp's OEIS frontend

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

Paul D. Hanna, Jun 01 2003

nonn

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

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

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

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

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

{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),", "))

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

A084613 a(n) = sum of absolute values of coefficients of (1 + x - 2*x^2)^n.

Original entry on oeis.org

1, 4, 14, 44, 124, 394, 1418, 4706, 14322, 40712, 135878, 468934, 1513650, 4502864, 13421408, 45258442, 152708520, 483810550, 1413811358, 4483843328, 15051967962, 49724234652, 154802614364, 461020649750, 1486736569982

Offset: 0

Paul D. Hanna, Jun 01 2003

nonn

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

Cf. A002426, A084600 to A084612, A084614, A084615.

A084612:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-2)^j: j in [0..k]]) >;
[(&+[Abs(A084612(n,k)): k in [0..2*n]]): n in [0..50]]; // G. C. Greubel, Mar 25 2023

Table[Total[Abs[CoefficientList[Expand[(1+x-2x^2)^n],x]]],{n,0,30}]  (* Harvey P. Dale, Apr 21 2011 *)

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

def A084612(n,k): return sum(binomial(n,k-j)*binomial(k-j,j)*(-2)^j for j in range(k+1))
def A084613(n): return sum(abs(A084612(n,k)) for k in range(2*n+1))
[A084613(n) for n in range(51)] # G. C. Greubel, Mar 25 2023

cp's OEIS Frontend

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

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