This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A144704 #38 Jul 13 2024 02:45:55
%S A144704 1,-4,-48,-320,-1792,-9216,-45056,-212992,-983040,-4456448,-19922944,
%T A144704 -88080384,-385875968,-1677721600,-7247757312,-31138512896,
%U A144704 -133143986176,-566935683072,-2405181685760,-10170482556928
%N A144704 a(n) = 4^n*(1-2*n).
%C A144704 With the n-th term of A000984 (C(2n,n)) as numerator, |a(n)| is the denominator of the probability that a random walk with steps of +-1 will return to the starting point for the first time after 2n steps. - _Shel Kaphan_, Jan 12 2023
%H A144704 G. C. Greubel, <a href="/A144704/b144704.txt">Table of n, a(n) for n = 0..1000</a>
%H A144704 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16).
%F A144704 G.f.: (1-12*x)/(1-4*x)^2.
%F A144704 From _Amiram Eldar_, Aug 05 2020: (Start)
%F A144704 Sum_{n>=0} 1/a(n) = 1 - arctanh(1/2)/2.
%F A144704 Sum_{n>=0} (-1)^(n+1)/a(n) = 1 + arctan(1/2)/2. (End)
%F A144704 E.g.f.: (1 - 8*x)*exp(4*x). - _G. C. Greubel_, Jun 16 2022
%F A144704 Sum_{n >= 1} x^(2*n-1)/a(n) = 1/4 * log((1 - x/2)/(1 + x/2)). Eldar's two summations above follow from this on setting x = 1 and x = i. - _Peter Bala_, Jul 08 2024
%t A144704 LinearRecurrence[{8,-16},{1,-4},30] (* _Harvey P. Dale_, Jun 12 2019 *)
%o A144704 (SageMath) [4^n*(1-2*n) for n in (0..30)] # _G. C. Greubel_, Jun 16 2022
%Y A144704 Hankel transform of A100320.
%Y A144704 Cf. A000302, A165747.
%K A144704 easy,sign
%O A144704 0,2
%A A144704 _Paul Barry_, Sep 19 2008