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 A271622 #27 Nov 17 2024 16:10:27
%S A271622 1,1,2,2,5,2,18,-19,115,-296,1115,-3632,12868,-44803,159577,-570455,
%T A271622 2059182,-7476086,27311129,-100274479,369888135,-1370063926,
%U A271622 5093782015,-19002596870,71109902844,-266855928791,1004045621663,-3786790876945
%N A271622 Expansion of -2/(x*sqrt(4*x+1)+x-2).
%H A271622 G. C. Greubel, <a href="/A271622/b271622.txt">Table of n, a(n) for n = 0..1000</a>
%F A271622 a(n) = 1 + Sum_{k=1..n-1} (-1)^(n-k-1)*k*binomial(2*n-3*k-1,n-k-1)/(n-k).
%F A271622 a(n) ~ (-1)^n * 2^(2*n+2) / (81*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 11 2016
%F A271622 D-finite with recurrence: (-n+1)*a(n) + 3*(-n+3)*a(n-1) + 2*(2*n-5)*a(n-2) + (n-1)*a(n-3) + 2*(2*n-5)*a(n-4) = 0. - _R. J. Mathar_, Apr 15 2016
%t A271622 Table[1 + (Sum[(k (-1)^(n - k - 1) Binomial[2 n - 3 k - 1, n - k - 1])/(n - k), {k, 1, n - 1}]), {n, 0, 27}] (* or *)
%t A271622 CoefficientList[Series[-2/(x Sqrt[4 x + 1] + x - 2), {x, 0, 27}], x] (* _Michael De Vlieger_, Apr 15 2016 *)
%o A271622 (Maxima)
%o A271622 a(n):=1+(sum(((-1)^(n-k-1)*k*binomial(2*n-3*k-1,n-k-1))/(n-k),k,1,n-1));
%o A271622 (PARI) my(x='x+O('x^99)); Vec(-2/(x*sqrt(4*x+1)+x-2)) \\ _Altug Alkan_, Apr 15 2016
%Y A271622 Cf. A000108.
%K A271622 sign,easy
%O A271622 0,3
%A A271622 _Vladimir Kruchinin_, Apr 11 2016