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G.f.: (1 - 6*x - sqrt(36*x^2 - 16*x + 1))/(2*x).
8

%I #35 Sep 08 2022 08:45:10

%S 1,7,56,497,4760,48174,507696,5516133,61363736,695540258,8004487568,

%T 93283238986,1098653880688,13056472392796,156371970692448,

%U 1885491757551213,22870028390806296,278862330338622618

%N G.f.: (1 - 6*x - sqrt(36*x^2 - 16*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 7^C(n+1,2). - _Philippe Deléham_, Feb 11 2009

%H Vincenzo Librandi, <a href="/A082305/b082305.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(n) = (1/n)*Sum_{k=0..n} 7^k*C(n, k)*C(n, k-1), a(0)=1.

%F 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

%F a(n) ~ sqrt(14+8*sqrt(7))*(8+2*sqrt(7))^n*(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 14 2012

%F 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

%t 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 *)

%o (PARI) a(n)=if(n<1,1,sum(k=0,n,7^k*binomial(n,k)*binomial(n,k-1))/n)

%o (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

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-6*x-Sqrt(36*x^2-16*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