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A082302 Expansion of g.f.: (1 - 5*x - sqrt(25*x^2 - 14*x + 1))/(2*x).

%I A082302 #48 Sep 08 2022 08:45:10
%S A082302 1,6,42,330,2814,25422,239442,2326434,23151030,234784662,2417832186,
%T A082302 25216231866,265796560302,2827138163550,30306009654690,
%U A082302 327081253546770,3551148743559270,38758882760119590,425024567305557450
%N A082302 Expansion of g.f.: (1 - 5*x - sqrt(25*x^2 - 14*x + 1))/(2*x).
%C A082302 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, for n > 0, a(n) = (1/n)*Sum_{k=0..n} (m+1)^k*C(n,k)*C(n,k-1).
%C A082302 Hankel transform is 6^C(n+1,2). - _Philippe Deléham_, Feb 11 2009
%H A082302 Vincenzo Librandi, <a href="/A082302/b082302.txt">Table of n, a(n) for n = 0..200</a>
%H A082302 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.
%H A082302 Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
%F A082302 Equals 6*A078018(n) for n > 0.
%F A082302 a(0)=1; for n > 0, a(n) = (1/n)*Sum_{k=0..n} 6^k*C(n, k)*C(n, k-1).
%F A082302 D-finite with recurrence: (n+1)*a(n) + 7*(1-2n)*a(n-1) + 25*(n-2)*a(n-2) = 0. - _R. J. Mathar_, Dec 08 2011
%F A082302 a(n) ~ sqrt(12 + 7*sqrt(6))*(7 + 2*sqrt(6))^n/(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 14 2012
%F A082302 a(n) = 6*hypergeom([1 - n, -n], [2], 6) for n > 0. - _Peter Luschny_, May 22 2017
%F A082302 G.f.: 1/(1 - 5*x - x/(1 - 5*x - x/(1 - 5*x - x/(1 - 5*x - x/(1 - ...))))), a continued fraction. - _Ilya Gutkovskiy_, Apr 04 2018
%p A082302 A082302_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p A082302 for w from 1 to n do a[w] := 6*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a,list)end: A082302_list(18); # _Peter Luschny_, May 19 2011
%p A082302 a := n -> `if`(n=0, 1, 6*hypergeom([1 - n, -n], [2], 6)):
%p A082302 seq(simplify(a(n)), n=0..18); # _Peter Luschny_, May 22 2017
%t A082302 Table[SeriesCoefficient[(1-5*x-Sqrt[25*x^2-14*x+1])/(2*x),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%o A082302 (PARI) a(n)=if(n<1,1,sum(k=0,n,6^k*binomial(n,k)*binomial(n,k-1))/n)
%o A082302 (PARI) x='x+O('x^99); Vec((1-5*x-(25*x^2-14*x+1)^(1/2))/(2*x)) \\ _Altug Alkan_, Apr 04 2018
%o A082302 (GAP) Concatenation([1],List([1..20],n->(1/n)*Sum([0..n],k->6^k*Binomial(n,k)*Binomial(n,k-1)))); # _Muniru A Asiru_, Apr 05 2018
%o A082302 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-5*x-Sqrt(25*x^2-14*x+1))/(2*x))); // _G. C. Greubel_, Aug 16 2018
%Y A082302 Cf. A006318, A047891.
%K A082302 nonn
%O A082302 0,2
%A A082302 _Benoit Cloitre_, May 10 2003