A126967 - OEIS

%I #20 Jan 22 2025 08:52:44

%S 1,0,-4,48,-624,9600,-175680,3790080,-95235840,2752081920,

%T -90328089600,3328103116800,-136191650918400,6131573025177600,

%U -301213549769932800,16030999766605824000,-918678402394841088000,56387623092958789632000,-3690023220507773140992000,256425697620583349354496000

%N Expansion of e.g.f.: sqrt(1+4*x)/(1+2*x).

%C A row of an array that is under investigation.

%H G. C. Greubel, <a href="/A126967/b126967.txt">Table of n, a(n) for n = 0..365</a>

%F 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 a(n) = (-2)^n*n!*JacobiP(n, -1/2, -(n+1), 3). - _Peter Luschny_, Jan 22 2025

%p seq(coeff(series( sqrt(1+4*x)/(1+2*x), x, n+1)*n!, x, n), n = 0..20);

%p # _G. C. Greubel_, Jan 29 2020

%p A126967 := n -> (-2)^n*n!*JacobiP(n, -1/2, -(n+1), 3):

%p seq(simplify(A126967(n)), n = 0..19);  # _Peter Luschny_, Jan 22 2025

%t nmax=20; CoefficientList[Series[Sqrt[1 + 4 x] / (1 + 2 x), {x, 0, nmax}], x] Range[0, nmax]! (* _Vincenzo Librandi_, Jan 24 2020 *)

%o (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 (PARI) my(x='x+O('x^30)); Vec(serlaplace( sqrt(1+4*x)/(1+2*x) )) \\ _G. C. Greubel_, Jan 29 2020

%o (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 Cf. A126966.

%K sign

%O 0,3

%A _N. J. A. Sloane_, Mar 22 2007