A116093 - OEIS

%I #30 Sep 08 2022 08:45:24

%S 1,-2,0,16,-56,48,384,-1920,3168,8512,-66560,161280,113920,-2224640,

%T 7311360,-3354624,-69253632,306754560,-408059904,-1898029056,

%U 12054196224,-25377005568,-38874316800,443400781824,-1289598418944,-52751204352,15086928789504,-58620595404800

%N Expansion of 1/sqrt(1+4*x+12*x^2).

%C Apart from signs identical this is to A098336. - _Joerg Arndt_, Jun 30 2013

%C Fourth binomial transform of the expansion of 1/sqrt(1-4*x+12*x^2), A098336.

%H Seiichi Manyama, <a href="/A116093/b116093.txt">Table of n, a(n) for n = 0..1000</a>

%H Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

%F E.g.f.: exp(-2*x)*Bessel_I(0,2*sqrt(-2)*x).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(n-k,k)(-2)^(n-k).

%F D-finite with recurrence: n*a(n) +2*(2*n-1)*a(n-1) +12*(n-1)*a(n-2)=0. - _R. J. Mathar_, Nov 07 2012

%F G.f.:  G(0), where G(k)= 1 - 2*x*(1+3*x)*(4*k+1)/( 2*k+1 - x*(1+3*x)*(2*k+1)*(4*k+3)/(x*(1+3*x)*(4*k+3) - (k+1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jun 30 2013

%t CoefficientList[Series[1/Sqrt[1+4x+12x^2],{x,0,30}],x] (* _Harvey P. Dale_, Oct 15 2014 *)

%o (PARI) my(x='x+O('x^30)); Vec(1/sqrt(1+4*x+12*x^2)) \\ _G. C. Greubel_, May 10 2019

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1+4*x+12*x^2) )); // _G. C. Greubel_, May 10 2019

%o (Sage) (1/sqrt(1+4*x+12*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 10 2019

%Y Column 2 of A307819.

%Y Cf. A098336.

%K easy,sign

%O 0,2

%A _Paul Barry_, Feb 04 2006