A098340 Expansion of 1/sqrt(1 - 6x + 21x^2).
1, 3, 3, -27, -189, -567, 189, 11259, 59859, 129033, -395847, -4730481, -19580211, -21264201, 258785523, 1917734373, 6051991059, -2659507911, -135544952151, -738957668337, -1618780564359, 5297724346923, 63513121347063, 266379249285873, 286776522444861, -3683959713627417
Offset: 0
Links
- Seiichi Manyama, 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.
Programs
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Maple
a := n -> 3^n*hypergeom([1/2 - n/2, -n/2], [1], -4/3): seq(simplify(a(n)), n=0..21); # Peter Luschny, Mar 19 2018
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Mathematica
CoefficientList[Series[1/Sqrt[1-6x+21x^2],{x,0,30}],x] (* Harvey P. Dale, Oct 07 2012 *) -
PARI
my(x = 'x + O('x^30)); Vec(1/sqrt(1-6*x+21*x^2)) \\ Jinyuan Wang, Sep 08 2019
Formula
E.g.f.: exp(3*x)*BesselI(0, 2*sqrt(-3)*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*3^n*(-3)^(-k).
a(n) = 3^n*Sum{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*(-3)^(-k).
D-finite with recurrence: n*a(n) + 3*(1-2*n)*a(n-1) + 21*(n-1)*a(n-2) = 0. - R. J. Mathar, Sep 26 2012
Lim sup n->infinity |a(n)|^(1/n) = sqrt(21). - Vaclav Kotesovec, Sep 29 2013
a(n) = 3^n*hypergeom([1/2 - n/2, -n/2], [1], -4/3). - Peter Luschny, Mar 19 2018
Comments