A082367 G.f.: (1-8*x-sqrt(64*x^2-20*x+1))/(2*x).

1, 9, 90, 981, 11430, 140058, 1782900, 23369805, 313426350, 4281280686, 59360821740, 833312907522, 11820849447420, 169182862497108, 2440064033240040, 35428651752626109, 517446157031236350

Offset: 0

Author

Benoit Cloitre, May 10 2003

Keywords

• nonn

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.

Crossrefs

Cf. A006318, A047891.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-8*x-Sqrt(64*x^2-20*x+1))/(2*x))); // G. C. Greubel, Sep 16 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 *)

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

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