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A098334
Expansion of 1/sqrt(1-2x+17x^2).
6
1, 1, -7, -23, 49, 401, 41, -5767, -11423, 65569, 299353, -441847, -5511791, -3665999, 79937417, 212712857, -861871423, -5076450239, 3966949049, 89482678313, 110424995569, -1233175514671, -4202194115863, 11830822055353, 91629438996001, -13485315511199
OFFSET
0,3
COMMENTS
Central coefficients of (1+x-4x^2)^n.
Binomial transform of 1/sqrt(1+16x^2), or (1,0,-8,0,96,0,-1280,...).
Binomial transform is A098337.
LINKS
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, 4*I*x), I=sqrt(-1);
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k, k)*(-4)^k;
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*(-4)^k;
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k, floor(k/2))*cos(Pi*k/2)*2^k.
D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +17*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
Lim sup n->oo |a(n)|^(1/n) = sqrt(17). - Vaclav Kotesovec, Feb 09 2014
a(n) = hypergeom([-n/2, 1/2-n/2], [1], -16). - Peter Luschny, Sep 18 2014
a(n) = (sqrt(17))^n*P(n,1/sqrt(17)), where P(n,x) is the Legendre polynomial of degree n. - Peter Bala, Mar 18 2018
MAPLE
a := n -> hypergeom([-n/2, 1/2-n/2], [1], -16);
seq(round(evalf(a(n), 99)), n=0..28); # Peter Luschny, Sep 18 2014
MATHEMATICA
CoefficientList[Series[1/Sqrt[1-2*x+17*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2014 *)
PROG
(PARI) x='x+O('x^99); Vec(1/(1-2*x+17*x^2)^(1/2)) \\ Altug Alkan, Mar 18 2018
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Sep 03 2004
STATUS
approved