A126967 Expansion of e.g.f.: sqrt(1+4*x)/(1+2*x).

1, 0, -4, 48, -624, 9600, -175680, 3790080, -95235840, 2752081920, -90328089600, 3328103116800, -136191650918400, 6131573025177600, -301213549769932800, 16030999766605824000, -918678402394841088000, 56387623092958789632000, -3690023220507773140992000, 256425697620583349354496000

Offset: 0

Author

N. J. A. Sloane, Mar 22 2007

Keywords

• sign

Comments

A row of an array that is under investigation.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..365

Crossrefs

Cf. A126966.

Programs

Magma

m:=20; R:=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

Maple

seq(coeff(series( sqrt(1+4*x)/(1+2*x), x, n+1)*n!, x, n), n = 0..20);
# G. C. Greubel, Jan 29 2020
A126967 := n -> (-2)^n*n!*JacobiP(n, -1/2, -(n+1), 3):
seq(simplify(A126967(n)), n = 0..19);  # Peter Luschny, Jan 22 2025

Mathematica

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

PARI

my(x='x+O('x^30)); Vec(serlaplace( sqrt(1+4*x)/(1+2*x) )) \\ G. C. Greubel, Jan 29 2020

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

Formula

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

a(n) = (-2)^n*n!*JacobiP(n, -1/2, -(n+1), 3). - Peter Luschny, Jan 22 2025