A082305 G.f.: (1 - 6*x - sqrt(36*x^2 - 16*x + 1))/(2*x).

1, 7, 56, 497, 4760, 48174, 507696, 5516133, 61363736, 695540258, 8004487568, 93283238986, 1098653880688, 13056472392796, 156371970692448, 1885491757551213, 22870028390806296, 278862330338622618

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 a(n) = (1/n)*Sum_{k=0..n} (m+1)^k * C(n,k) *C(n,k-1) for n > 0.

Hankel transform is 7^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:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-6*x-Sqrt(36*x^2-16*x+1))/(2*x))); // G. C. Greubel, Sep 16 2018

Mathematica

Table[SeriesCoefficient[(1-6*x-Sqrt[36*x^2-16*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,7^k*binomial(n,k)*binomial(n,k-1))/n)

PARI

x='x+O('x^99); Vec((1-6*x-(36*x^2-16*x+1)^(1/2))/(2*x)) \\ Altug Alkan, Apr 04 2018

Formula

a(n) = (1/n)*Sum_{k=0..n} 7^k*C(n, k)*C(n, k-1), a(0)=1.

D-finite with recurrence: (n+1)*a(n) + 8*(1-2*n)*a(n-1) + 36*(n-2)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011

a(n) ~ sqrt(14+8*sqrt(7))*(8+2*sqrt(7))^n*(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012

G.f.: 1/(1 - 6*x - x/(1 - 6*x - x/(1 - 6*x - x/(1 - 6*x - x/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Apr 04 2018