A116092 - OEIS

%I #18 Sep 08 2022 08:45:24

%S 1,-4,-8,224,-1184,-2944,84736,-467968,-1235456,35956736,-202108928,

%T -548651008,16063381504,-91151859712,-251452325888,7389369073664,

%U -42180470767616,-117581870006272,3464100777558016,-19854347412176896,-55753417460547584,1645577388148391936

%N Expansion of 1/sqrt(1+8*x+64*x^2).

%C 8th binomial transform is expansion of 1/sqrt(1-8*x+64*x^2).

%H G. C. Greubel, <a href="/A116092/b116092.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(-4*x)*Bessel_I(0, 2*sqrt(-12)*x).

%F a(n) = 2^n*Sum_{k=0..n} C(n,n-k)*C(n,k)*(-3)^k.

%F a(n) = 2^n*A116091(n).

%F D-finite with recurrence: n*a(n) +4*(2*n-1)*a(n-1) +64*(n-1)*a(n-2)=0. - _R. J. Mathar_, Nov 07 2012

%t CoefficientList[Series[1/Sqrt[1+8*x+64*x^2], {x, 0, 30}], x] (* _G. C. Greubel_, May 10 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec(1/sqrt(1+8*x+64*x^2)) \\ _G. C. Greubel_, May 10 2019

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1+8*x+64*x^2) )); // _G. C. Greubel_, May 10 2019

%o (Sage) (1/sqrt(1+8*x+64*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 10 2019

%o (GAP) List([0..30], n-> 2^n*Sum([0..n], k-> (-3)^k*Binomial(n,k)* Binomial(n, n-k))) # _G. C. Greubel_, May 10 2019

%K easy,sign

%O 0,2

%A _Paul Barry_, Feb 04 2006