A098335 - OEIS

%I #40 Jan 30 2020 21:29:15

%S 1,2,2,-4,-26,-68,-76,184,1222,3308,3772,-9656,-64676,-177448,-203992,

%T 536176,3607622,9968972,11510636,-30723416,-207302156,-575382392,

%U -666187432,1796105744,12142184476,33803271032

%N Expansion of 1/sqrt(1-4x+8x^2).

%C Central coefficients of (1+2x-x^2)^n. Binomial transform of A098331.

%C Diagonal of rational function 1/(1 - (x^2 + 2*x*y - y^2)). - _Gheorghe Coserea_, Aug 04 2018

%H G. C. Greubel, <a href="/A098335/b098335.txt">Table of n, a(n) for n = 0..1000</a>

%H Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

%F From _Paul Barry_, Sep 08 2004: (Start)

%F E.g.f. : exp(2*x)*BesselI(0, 2*I*x), I=sqrt(-1);

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(n-k,k)*2^n*(-4)^(-k).

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

%F E.g.f. : exp(2*x)*BesselJ(0, 2*x). - _Sergei N. Gladkovskii_, Aug 22 2012

%F It appears that a(j+2) = (2*(2*j+1)*a(j+1))/(j+1)-(8*j*a(j))/(j+1), in case of re-indexing from 0 to 1. - _Alexander R. Povolotsky_, Aug 22 2012

%F D-finite with recurrence: a(n+2) = ((4*n+6)*a(n+1) - 8*(n+1)*a(n))/(n+2); a(0)=1,a(1)=2. - _Sergei N. Gladkovskii_, Aug 22 2012

%F a(n) = 2^n*_2F_1(1/2-n/2, -n/2, 1, -1), where _2F_1(a,b;c;x) is the hypergeometric function. - _Alexander R. Povolotsky_, Aug 22 2012

%t a[n_] := 2^n*Hypergeometric2F1[1/2 - n/2, -n/2, 1, -1]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Sep 05 2012, after _Alexander R. Povolotsky_ *)

%t CoefficientList[Series[1/Sqrt[1 - 4*x + 8*x^2], {x,0,50}], x] (* _G. C. Greubel_, Feb 19 2017 *)

%o (PARI) x='x+O('x^25); Vec(1/sqrt(1 - 4*x + 8*x^2)) \\ _G. C. Greubel_, Feb 19 2017

%K easy,sign

%O 0,2

%A _Paul Barry_, Sep 03 2004