A098441 Expansion of 1/sqrt(1 - 2*x - 63*x^2).

1, 1, 33, 97, 1729, 8001, 105441, 627873, 6989697, 48363649, 488206753, 3701949153, 35289342529, 283146701761, 2610495177057, 21695983405857, 196218339243777, 1667338615773441, 14917038493453089, 128562758660255073

Offset

0,3

Comments

Binomial transform of 1/sqrt(1-64*x^2).

It appears that a(n) is the coefficient of x^n in (x^2+x+16)^n. - Joerg Arndt, Jan 13 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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, 4*sqrt(4)*x).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*binomial(n, k)*16^k.

D-finite with recurrence: n*a(n) = (2*n-1)*a(n-1) + 63*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012

a(n) ~ 3^(2*n+1)/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012

a(n) = 4^n*GegenbauerC(n, -n, -1/8). - Peter Luschny, May 08 2016

a(n) = hypergeometric2F1((1 - n)/2, -n/2, 1, 64). - G. C. Greubel, Feb 21 2017

Maple

a := n -> simplify(4^n*GegenbauerC(n, -n, -1/8)):
seq(a(n), n=0..19); # Peter Luschny, May 08 2016

Mathematica

Table[SeriesCoefficient[1/Sqrt[1-2*x-63*x^2], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 15 2012 *)
CoefficientList[Series[1/Sqrt[1-2x-63x^2], {x, 0, 30}], x] (* Harvey P. Dale, Jan 27 2017 *)
Table[Hypergeometric2F1[(1 - k)/2, -k/2, 1, 64], {k, 0, 50}] (* G. C. Greubel, Feb 21 2017 *)

Prog

(PARI) x='x+O('x^66); Vec(1/sqrt(1-2*x-63*x^2)) \\ Joerg Arndt, May 11 2013
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/Sqrt(1-2*x-63*x^2))); // G. C. Greubel, Oct 08 2018

Crossrefs

Sequence in context: A044220 A044601 A195315 * A032650 A284986 A133901

Adjacent sequences: A098438 A098439 A098440 * A098442 A098443 A098444

Keyword

easy,nonn

Author

Paul Barry, Sep 07 2004

Status

approved