A082366 G.f.: (1 - 7*x - sqrt(49*x^2 - 18*x + 1))/(2*x).

1, 8, 72, 712, 7560, 84616, 985032, 11814728, 145043208, 1813915912, 23029334856, 296050614216, 3846007927944, 50412893051784, 665925356663496, 8855844075949128, 118467982501096968, 1593108078166843912

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 8^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

GAP

Concatenation([1],List([1..20],n->(1/n)*Sum([0..n],k->8^k*Binomial(n,k)*Binomial(n,k-1)))); # Muniru A Asiru, Apr 05 2018

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-7*x-Sqrt(49*x^2-18*x+1))/(2*x))); // G. C. Greubel, Sep 16 2018

Mathematica

CoefficientList[Series[(1-7x-Sqrt[49x^2-18x+1])/(2x),{x,0,20}],x]  (* Harvey P. Dale, Feb 22 2011 *)

PARI

a(n)=if(n<1,1,sum(k=0,n,8^k*binomial(n,k)*binomial(n,k-1))/n)

PARI

x='x+O('x^99); Vec((1-7*x-(49*x^2-18*x+1)^(1/2))/(2*x)) \\ Altug Alkan, Apr 04 2018

Formula

a(0)=1; a(n) = (1/n)*Sum_{k=0..n} 8^k*C(n, k)*C(n, k-1) for n > 0.

D-finite with recurrence: (n+1)*a(n) + 9*(1-2n)*a(n-1) + 49*(n-2)*a(n-2) = 0. - R. J. Mathar, Dec 08 2011

a(n) ~ sqrt(16+18*sqrt(2))*(9+4*sqrt(2))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012

G.f.: 1/(1 - 7*x - x/(1 - 7*x - x/(1 - 7*x - x/(1 - 7*x - x/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Apr 04 2018