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

Author

Paul Barry, Sep 07 2004

Keywords

• easy
• nonn

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.
• Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See pp. 96-97.

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1/Sqrt(1-2*x-63*x^2))); // G. C. Greubel, Oct 08 2018

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 *)

PARI

x='x+O('x^66); Vec(1/sqrt(1-2*x-63*x^2)) \\ Joerg Arndt, May 11 2013

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