OFFSET
0,2
COMMENTS
More generally coefficients of (1-m*x-sqrt(m^2*x^2-(2*m+4)*x+1))/(2*x) are given by a(0)=1 and n>0 a(n)=(1/n)*Sum_{k=0..n} (m+1)^k*C(n,k)*C(n,k-1).
Hankel transform is 9^C(n+1,2). - Philippe Deléham, Feb 11 2009
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
FORMULA
a(0)=1; for n > 0, a(n) = (1/n)*Sum_{k=0..n} 9^k*C(n, k)*C(n, k-1).
D-finite with recurrence: (n+1)*a(n) + 10*(1-2n)*a(n-1) + 64*(n-2)*a(n-2) = 0. - R. J. Mathar, Dec 08 2011 Recurrence follows from the D.E. (x-20*x^2+64*x^3)*y' + (1-10*x)*y - 1 - 8*x = 0 satisfied by the g.f.. - Robert Israel, Mar 16 2018
a(n) ~ sqrt(3)*2^(4*n+1)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
G.f.: 1/(1 - 8*x - x/(1 - 8*x - x/(1 - 8*x - x/(1 - 8*x - x/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Apr 04 2018
MAPLE
f:= gfun:-rectoproc({64*n*a(n)+(-30-20*n)*a(1+n)+(3+n)*a(n+2), a(0) = 1, a(1) = 9}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 16 2018
MATHEMATICA
Table[SeriesCoefficient[(1-8*x-Sqrt[64*x^2-20*x+1])/(2*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)
PROG
(PARI) a(n)=if(n<1, 1, sum(k=0, n, 9^k*binomial(n, k)*binomial(n, k-1))/n)
(PARI) x='x+O('x^99); Vec((1-8*x-(64*x^2-20*x+1)^(1/2))/(2*x)) \\ Altug Alkan, Apr 04 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-8*x-Sqrt(64*x^2-20*x+1))/(2*x))); // G. C. Greubel, Sep 16 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, May 10 2003
STATUS
approved