A098332 Expansion of 1/sqrt(1 - 2*x + 9*x^2).

1, 1, -3, -11, 1, 81, 141, -363, -1791, -479, 13597, 29877, -54911, -353807, -223443, 2539989, 6806529, -8302527, -73999299, -73313931, 489731841, 1584548241, -1110170163, -15812965611, -21391839999, 94696016481

Offset

0,3

Comments

Central coefficients of (1 + x - 2*x^2)^n.

Binomial transform of 1/sqrt(1+8*x^2), or (1,0,-4,0,24,0,...).

Binomial transform is A098336.

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, ex. 7.56, p. 575.

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.

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Formula

E.g.f.: exp(x)*BesselI(0, 2*sqrt(-2)*x);

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

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

a(n) = (-1)^n * Sum_{k=0..n} binomial(n,k)^2 * (-2)^k.

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

D-finite with recurrence: a(n+2) = ((2*n+3)*a(n+1))/(n+2) - (9*(n+1)*a(n))/(n+2) with a(0)=1, a(1)=1. (See Graham, Knuth, and Patashnik). - Alexander R. Povolotsky, Aug 23 2012

a(n) = hypergeom([1/2-n/2, -n/2], [1], -8). - Peter Luschny, Sep 18 2014

a(n) = (3/2)*(9/2)^n*Sum_{k >= 0} (-1/2)^k*binomial(n+k,k)^2. - Peter Bala, Mar 02 2017

Maple

a := n -> hypergeom([1/2-n/2, -n/2], [1], -8);
seq(round(evalf(a(n), 99)), n=0..30); # Peter Luschny, Sep 18 2014

Mathematica

Table[(-3)^n*LegendreP[n, -1/3], {n, 0, 20}] (* Vaclav Kotesovec, Jul 23 2013 *)
CoefficientList[Series[1/Sqrt[1 - 2*x + 9*x^2], {x, 0, 50}], x] (* G. C. Greubel, Feb 18 2017 *)

Prog

(PARI) x='x+O('x^25); Vec(1/sqrt(1 - 2*x + 9*x^2)) \\ G. C. Greubel, Feb 18 2017

Crossrefs

Cf. A126869, A012000, A116091, A098341.

Sequence in context: A092528 A069604 A354979 * A096663 A302120 A133369

Adjacent sequences: A098329 A098330 A098331 * A098333 A098334 A098335

Keyword

easy,sign

Author

Paul Barry, Sep 03 2004

Status

approved