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%I #16 Sep 08 2022 08:45:23
%S 1,6,21,60,156,384,912,2112,4800,10752,23808,52224,113664,245760,
%T 528384,1130496,2408448,5111808,10813440,22806528,47972352,100663296,
%U 210763776,440401920,918552576,1912602624,3976200192,8254390272,17112760320
%N Expansion of ((1+x)/(1-2x))^2.
%C Binomial transform is A014915. In general, ((1+x)/(1-r*x))^2 expands to a(n) = ((r+1)*r^n*((r+1)*n + r - 1) + 0^n)/r^2, which is also a(n) = Sum_{k=0..n} C(n,k)*Sum_{j=0..k} (j+1)*(r+1)^j. This is the self-convolution of the coordination sequence for the infinite tree with valency r.
%H Vincenzo Librandi, <a href="/A113070/b113070.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 (4,-4).
%F G.f.: (1+x)^2/(1-2x)^2;
%F a(n) = 3*2^n(3n+1)/4 + 0^n/4;
%F a(n) = Sum_{k=0..n} A003945(k)*A003945(n-k);
%F a(n) = Sum_{k=0..n} C(n, k)*Sum_{j=0..k} (j+1)*3^j.
%F a(n) = 4*a(n-1) - 4*a(n-2); a(0)=1, a(1)=6, a(2)=21. - _Harvey P. Dale_, May 20 2011
%t Join[{1},LinearRecurrence[{4,-4},{6,21},30]] (* or *) CoefficientList[ Series[((1+x)/(1-2x))^2,{x,0,30}],x] (* _Harvey P. Dale_, May 20 2011 *)
%o (Magma) [3*2^n*(3*n+1)/4+0^n/4: n in [0..30]]; // _Vincenzo Librandi_, May 21 2011
%Y Cf. A113071.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Oct 14 2005
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