A116091 Expansion of 1/sqrt(1+4*x+16*x^2).

1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228, -197372, -267896, 3921724, -11126936, -15347432, 225505648, -643622906, -897078476, 13214495764, -37869162392, -53170602284, 784672445368, -2255295815192, -3183829452272, 47051201187676, -135537088268792, -192142210448216

Offset

0,2

Comments

Fourth binomial transform is expansion of 1/sqrt(1-4*x+16*x^2), A012000.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Hacène Belbachir, Abdelghani Mehdaoui, and 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(-2*x)*Bessel_I(0, 2*sqrt(-3)*x).

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

O.g.f.: P(-1/2,4*x) with the o.g.f. P(x,z):=1/sqrt(1-2*x*z+z^2) for the Legendre polynomials. Wolfdieter Lang, Mar 10 2011.

G.f. A(x) = 1/(2*T(0)-4*x-1) where T(k)= 1 + 3*x/(1 - x/T(k+1)); (continued fraction, 2-step ). - Sergei N. Gladkovskii, Aug 23 2012

D-finite with recurrence: a(n+2) = -(16*(n+1)*a(n))/(n+2) - (2*(2*n+3)*a(n+1))/(n+2) with a(0)=1, a(1)=-2. - Alexander R. Povolotsky, Aug 23 2012

a(n) = (-4)^n*hypergeom([-n, 1+n], [1], 1/4). - Peter Luschny, May 09 2016

From Peter Bala, Nov 30 2021: (Start)

a(n) = (-4)^n^P(n,1/2), where P(n,x) is the n-th Legendre polynomial.

a(n) = (4/3)*(16^n)*Sum_{k >= n} C(k,n)^2*(-1/3)^k.

a(n) = (-3)^n*hypergeom([-n, -n], [1], -1/3).

a(n) = (4/3)*(-16/3)^n*hypergeom([n+1, n+1], [1], -1/3).

a(n) = [x^n] ((1 + x)*(3 - x))^n. (End)

Maple

a := n -> (-4)^n*hypergeom([-n, 1+n], [1], 1/4);
seq(simplify(a(n)), n=0..26); # Peter Luschny, May 09 2016

Mathematica

Table[4^n*LegendreP[n, -1/2], {n, 0, 30}] (* Vaclav Kotesovec, Jul 23 2013 *)
CoefficientList[Series[1/Sqrt[1+4x+16x^2], {x, 0, 30}], x] (* Harvey P. Dale, Jun 08 2015 *)

Prog

(PARI) Vec(1/sqrt(1+4*x+16*x^2+O(x^30))) \\ M. F. Hasler, Aug 25 2012
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1+4*x+16*x^2) )); // G. C. Greubel, May 09 2019
(Sage) (1/sqrt(1+4*x+16*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

Crossrefs

Cf. A012000, A098332, A126869.

Sequence in context: A246062 A178955 A012000 * A127262 A359998 A343172

Adjacent sequences: A116088 A116089 A116090 * A116092 A116093 A116094

Keyword

sign,easy

Author

Paul Barry, Feb 04 2006

Status

approved