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A241204 Expansion of (1 + 2*x)^2/(1 - 2*x)^2.

%I A241204 #42 Jun 23 2025 12:23:44
%S A241204 1,8,32,96,256,640,1536,3584,8192,18432,40960,90112,196608,425984,
%T A241204 917504,1966080,4194304,8912896,18874368,39845888,83886080,176160768,
%U A241204 369098752,771751936,1610612736,3355443200,6979321856,14495514624,30064771072,62277025792
%N A241204 Expansion of (1 + 2*x)^2/(1 - 2*x)^2.
%H A241204 Vincenzo Librandi, <a href="/A241204/b241204.txt">Table of n, a(n) for n = 0..1000</a>
%H A241204 Sela Fried, <a href="https://arxiv.org/abs/2406.18923">Counting r X s rectangles in nondecreasing and Smirnov words</a>, arXiv:2406.18923 [math.CO], 2024. See p. 12.
%H A241204 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).
%F A241204 a(n) = 2^(2+n)*n for n>0. - _Colin Barker_, Apr 23 2014
%F A241204 a(n) = 4*a(n-1)-4*a(n-2) for n>2. - _Colin Barker_, Apr 23 2014
%F A241204 From _Amiram Eldar_, Jan 13 2021: (Start)
%F A241204 Sum_{n>=1} 1/a(n) = log(2)/4.
%F A241204 Sum_{n>=1} (-1)^(n+1)/a(n) = log(3/2)/4. (End)
%F A241204 E.g.f.: 1 + 8*x*exp(x). - _G. C. Greubel_, Jun 07 2023
%p A241204 A241204:= n->`if`(n=0, 1, 2^(n+2)*n); seq(A241204(n), n=0..20); # _Wesley Ivan Hurt_, Apr 22 2014
%t A241204 Table[2^(n+2)*n + Boole[n==0], {n,0,40}] (* _G. C. Greubel_, Jun 07 2023 *)
%t A241204 LinearRecurrence[{4,-4},{1,8,32},30] (* _Harvey P. Dale_, Jun 23 2025 *)
%o A241204 (Magma) R<x>:=PowerSeriesRing(Integers(), 41); Coefficients(R!((1+2*x)^2/(1-2*x)^2));
%o A241204 (PARI) Vec((2*x+1)^2/(2*x-1)^2 + O(x^100)) \\ _Colin Barker_, Apr 22 2014
%o A241204 (Sage)
%o A241204 def A241204(i):
%o A241204     if i==0: return 1
%o A241204     else: return 2^(2+i)*i;
%o A241204 [A241204(n) for n in (0..30)] # _Bruno Berselli_, Apr 23 2014
%Y A241204 Subsequence of A008574.
%K A241204 nonn,easy
%O A241204 0,2
%A A241204 _Ilya Lopatin_ and _Juri-Stepan Gerasimov_, Apr 17 2014