A104721 - OEIS

%I #19 Sep 08 2022 08:45:17

%S 1,2,5,8,20,32,80,128,320,512,1280,2048,5120,8192,20480,32768,81920,

%T 131072,327680,524288,1310720,2097152,5242880,8388608,20971520,

%U 33554432,83886080,134217728,335544320,536870912,1342177280,2147483648

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

%C Binomial transform is A033113.

%C 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).

%C 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

%H G. C. Greubel, <a href="/A104721/b104721.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,4).

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

%t 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 *)

%o (PARI) vector(40, n, n--; (9*2^n +(-2)^n -2*0^n)/8) \\ _G. C. Greubel_, Jul 14 2019

%o (Magma) [1] cat [9*2^(n-3) -(-2)^(n-3): n in [1..40]]; // _G. C. Greubel_, Jul 14 2019

%o (Sage) [1]+[9*2^(n-3) -(-2)^(n-3) for n in (1..40)] # _G. C. Greubel_, Jul 14 2019

%o (GAP) Concatenation([1], List([1..40], n-> 9*2^(n-3) -(-2)^(n-3))); # _G. C. Greubel_, Jul 14 2019

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 20 2005