A104721 Expansion of (1+x)^2/(1-4*x^2).

1, 2, 5, 8, 20, 32, 80, 128, 320, 512, 1280, 2048, 5120, 8192, 20480, 32768, 81920, 131072, 327680, 524288, 1310720, 2097152, 5242880, 8388608, 20971520, 33554432, 83886080, 134217728, 335544320, 536870912, 1342177280, 2147483648

Offset: 0

Author

Paul Barry, Mar 20 2005

Keywords

• easy
• nonn

Comments

Binomial transform is A033113.

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

If a(1)=2 is dropped, sequence becomes identical to A084568 (Proof immediate by standard manipulation of the two generating functions). - R. J. Mathar, May 19 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,4).

Programs

GAP

Concatenation([1], List([1..40], n-> 9*2^(n-3) -(-2)^(n-3))); # G. C. Greubel, Jul 14 2019

Magma

[1] cat [9*2^(n-3) -(-2)^(n-3): n in [1..40]]; // G. C. Greubel, Jul 14 2019

Mathematica

CoefficientList[Series[(1+x)^2/(1-4x^2),{x,0,40}],x] (* or *) LinearRecurrence[{0,4},{1,2,5},40] (* Harvey P. Dale, Dec 05 2015 *)

PARI

vector(40, n, n--; (9*2^n +(-2)^n -2*0^n)/8) \\ G. C. Greubel, Jul 14 2019

Sage

[1]+[9*2^(n-3) -(-2)^(n-3) for n in (1..40)] # G. C. Greubel, Jul 14 2019

Formula

a(n) = (9*2^n + (-2)^n - 2*0^n)/8.