A116092 Expansion of 1/sqrt(1+8*x+64*x^2).

1, -4, -8, 224, -1184, -2944, 84736, -467968, -1235456, 35956736, -202108928, -548651008, 16063381504, -91151859712, -251452325888, 7389369073664, -42180470767616, -117581870006272, 3464100777558016, -19854347412176896, -55753417460547584, 1645577388148391936

Offset: 0

Author

Paul Barry, Feb 04 2006

Keywords

• easy
• sign

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Crossrefs

Cf. A116091.

Programs

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

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1+8*x+64*x^2) )); // G. C. Greubel, May 10 2019

Mathematica

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

PARI

my(x='x+O('x^30)); Vec(1/sqrt(1+8*x+64*x^2)) \\ G. C. Greubel, May 10 2019

Sage

(1/sqrt(1+8*x+64*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 10 2019

Formula

E.g.f.: exp(-4*x)*Bessel_I(0, 2*sqrt(-12)*x).

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

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

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