A098337 Expansion of 1/sqrt(1-4x+20x^2).

1, 2, -4, -40, -80, 352, 2624, 3712, -32000, -186880, -134144, 2885632, 13520896, -1269760, -256000000, -966164480, 1056112640, 22286827520, 66722201600, -162411315200, -1901125959680, -4329895362560

Offset: 0

Author

Paul Barry, Sep 03 2004

Keywords

• easy
• sign

Comments

Central coefficients of (1+2x-4x^2)^n. Binomial transform of A098334.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Hacène Belbachir, Abdelghani Mehdaoui, 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.

Programs

Mathematica

Table[Sum[Binomial[n,k]*Binomial[2*(n-k), n]*(-5)^k,{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Feb 08 2014 *)
CoefficientList[Series[1/Sqrt[1-4x+20x^2],{x,0,30}],x] (* Harvey P. Dale, Jul 29 2015 *)

Formula

E.g.f.: exp(2x)BesselI(0, 4*I*x), I=sqrt(-1);

a(n) = sum{k=0..floor(n/2), binomial(n, k)binomial(n-k, k)2^n(-1)^k};

a(n) = sum{k=0..n, binomial(2k, k)binomial(k, n-k)(-5)^(n-k)}.

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)binomial(2(n-k), n)(-5)^k. - Paul Barry, Sep 08 2004.

D-finite with recurrence: n*a(n) +2*(-2*n+1)*a(n-1) +20*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 24 2012

Lim sup n->infinity |a(n)|^(1/n) = 2*sqrt(5). - Vaclav Kotesovec, Feb 08 2014