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

0,2

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.

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

Mathematica

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

Prog

(PARI) my(x='x+O('x^30)); Vec(1/sqrt(1+8*x+64*x^2)) \\ G. C. Greubel, May 10 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 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
(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

Crossrefs

Sequence in context: A013065 A013096 A200729 * A085631 A074073 A090653

Adjacent sequences: A116089 A116090 A116091 * A116093 A116094 A116095

Keyword

easy,sign

Author

Paul Barry, Feb 04 2006

Status

approved