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 A126967 #20 Jan 22 2025 08:52:44
%S A126967 1,0,-4,48,-624,9600,-175680,3790080,-95235840,2752081920,
%T A126967 -90328089600,3328103116800,-136191650918400,6131573025177600,
%U A126967 -301213549769932800,16030999766605824000,-918678402394841088000,56387623092958789632000,-3690023220507773140992000,256425697620583349354496000
%N A126967 Expansion of e.g.f.: sqrt(1+4*x)/(1+2*x).
%C A126967 A row of an array that is under investigation.
%H A126967 G. C. Greubel, <a href="/A126967/b126967.txt">Table of n, a(n) for n = 0..365</a>
%F A126967 D-finite with recurrence: a(n) +6*(n-1)*a(n-1) +4*(n-1)*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Jan 23 2020
%F A126967 a(n) = (-2)^n*n!*JacobiP(n, -1/2, -(n+1), 3). - _Peter Luschny_, Jan 22 2025
%p A126967 seq(coeff(series( sqrt(1+4*x)/(1+2*x), x, n+1)*n!, x, n), n = 0..20);
%p A126967 # _G. C. Greubel_, Jan 29 2020
%p A126967 A126967 := n -> (-2)^n*n!*JacobiP(n, -1/2, -(n+1), 3):
%p A126967 seq(simplify(A126967(n)), n = 0..19); # _Peter Luschny_, Jan 22 2025
%t A126967 nmax=20; CoefficientList[Series[Sqrt[1 + 4 x] / (1 + 2 x), {x, 0, nmax}], x] Range[0, nmax]! (* _Vincenzo Librandi_, Jan 24 2020 *)
%o A126967 (Magma) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Sqrt(1+4*x)/(1+2*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _Vincenzo Librandi_, Jan 24 2020
%o A126967 (PARI) my(x='x+O('x^30)); Vec(serlaplace( sqrt(1+4*x)/(1+2*x) )) \\ _G. C. Greubel_, Jan 29 2020
%o A126967 (Sage) [factorial(n)*( sqrt(1+4*x)/(1+2*x) ).series(x,n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Jan 29 2020
%Y A126967 Cf. A126966.
%K A126967 sign
%O A126967 0,3
%A A126967 _N. J. A. Sloane_, Mar 22 2007