A097064 Expansion of (1 - 4*x + 6*x^2)/(1 - 2*x)^2.

1, 0, 2, 8, 24, 64, 160, 384, 896, 2048, 4608, 10240, 22528, 49152, 106496, 229376, 491520, 1048576, 2228224, 4718592, 9961472, 20971520, 44040192, 92274688, 192937984, 402653184, 838860800, 1744830464, 3623878656, 7516192768, 15569256448, 32212254720, 66571993088

Offset: 0

Author

Paul Barry, Jul 22 2004

Keywords

• easy
• nonn

Comments

Binomial transform of A097062.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-4).

Crossrefs

Essentially the same as A036289.

Cf. A001787, A002162, A016578, A097062.

Programs

Mathematica

CoefficientList[Series[(1-4x+6x^2)/(1-2x)^2,{x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{4,-4},{0,2},30]] (* Harvey P. Dale, May 26 2011 *)

Formula

a(n) = (n-1)*2^(n-1) + 3*0^n/2.

a(n) = 4*a(n-1) - 4*a(n-2), n>2.

a(n) = Sum_{k=0..n} binomial(n, k)*((2k-1)/2 + 3*(-1)^k/2).

a(n+1)/2 = A001787(n).

From Amiram Eldar, Oct 01 2022: (Start)

Sum_{n>=2} 1/a(n) = log(2) (A002162).

Sum_{n>=2} (-1)^n/a(n) = log(3/2) (A016578). (End)

E.g.f.: (3 - exp(2*x)*(1 - 2*x))/2. - Stefano Spezia, Feb 12 2023