%I #40 Sep 08 2022 08:45:10
%S 1,8,72,712,7560,84616,985032,11814728,145043208,1813915912,
%T 23029334856,296050614216,3846007927944,50412893051784,
%U 665925356663496,8855844075949128,118467982501096968,1593108078166843912
%N G.f.: (1 - 7*x - sqrt(49*x^2 - 18*x + 1))/(2*x).
%C 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.
%C Hankel transform is 8^C(n+1,2). - _Philippe Deléham_, Feb 11 2009
%H Vincenzo Librandi, <a href="/A082366/b082366.txt">Table of n, a(n) for n = 0..200</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%F a(0)=1; a(n) = (1/n)*Sum_{k=0..n} 8^k*C(n, k)*C(n, k-1) for n > 0.
%F 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
%F a(n) ~ sqrt(16+18*sqrt(2))*(9+4*sqrt(2))^n/(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 14 2012
%F 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
%t CoefficientList[Series[(1-7x-Sqrt[49x^2-18x+1])/(2x),{x,0,20}],x] (* _Harvey P. Dale_, Feb 22 2011 *)
%o (PARI) a(n)=if(n<1,1,sum(k=0,n,8^k*binomial(n,k)*binomial(n,k-1))/n)
%o (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
%o (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
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-7*x-Sqrt(49*x^2-18*x+1))/(2*x))); // _G. C. Greubel_, Sep 16 2018
%Y Cf. A006318, A047891.
%K nonn
%O 0,2
%A _Benoit Cloitre_, May 10 2003