A073388 - OEIS

%I #24 Oct 04 2022 08:38:32

%S 1,4,16,56,188,608,1920,5952,18192,54976,164608,489088,1443776,

%T 4238336,12382208,36022272,104407296,301618176,868765696,2495715328,

%U 7152286720,20452548608,58369409024

%N Convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

%H Muniru A Asiru, <a href="/A073388/b073388.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-8,-4).

%F a(n) = Sum_{k=0..n} b(k)*b(n-k), with b(k) = A002605(k).

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

%F a(n) = ((n+1)*U(n+1) + 2*(n+2)*U(n))/6, with U(n) = A002605(n), n >= 0.

%F G.f.: 1/(1-2*x*(1+x))^2.

%F a(n) = Sum_{k=0..floor((n+2)/2)} k*binomial(n-k+2, k)2^(n-k+1). - _Paul Barry_, Oct 15 2004

%t CoefficientList[Series[1/(1-2*x-2*x^2)^2, {x,0,40}], x] (* _G. C. Greubel_, Oct 03 2022 *)

%o (Sage) taylor( 1/(1-2*x-2*x^2)^2, x, 0, 24).list() # _Zerinvary Lajos_, Jun 03 2009; modified by _G. C. Greubel_, Oct 03 2022

%o (GAP) List([0..25], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+1,1)*Binomial(n-k,k)*(1/2)^k)); # _Muniru A Asiru_, Jun 12 2018

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^2 )); // _G. C. Greubel_, Oct 03 2022

%Y Second (m=1) column of triangle A073387.

%Y Cf. A002605.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 02 2002