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%I #16 Sep 22 2018 03:15:12
%S 1,1,1,5,19,79,333,1441,6351,28451,129185,593373,2752427,12876343,
%T 60684533,287857209,1373286375,6584979659,31719337353,153416338549,
%U 744777567043,3627787084319
%N Sequence such that the Hankel transform of a(n+1) satisfies a generalized Somos-4 recurrence.
%C The Hankel transform of a(n+1) satisfies a generalized Somos-4 Hankel determinant recurrence.
%C Hankel transform of a(n+1) is A160703. In general, we can conjecture that the Hankel transform of
%C h(n) of a(n+1), where a(n)=if(n=0,1,if(n=1,1,if(n=2,1,r*a(n-1)+s*sum{k=0..n-2, a(k)*a(n-1-k)}))),
%C satisfies the generalized Somos-4 recurrence
%C h(n)=(s^2*h(n-1)*h(n-3)+s^3*(2*s+r-2)*h(n-2)^2)/h(n-4).
%C The case r=s=1 is proved in the Xin reference.
%H Vincenzo Librandi, <a href="/A160702/b160702.txt">Table of n, a(n) for n = 0..300</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry5/barry223.html">On the Hurwitz Transform of Sequences</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
%H Gouce Xin, <a href="https://doi.org/10.1016/j.aam.2008.04.003">Proof of the Somos-4 Hankel determinants conjecture</a>, Advances in Applied Mathematics, Volume 42, Issue 2, February 2009, Pages 152-156.
%F a(n) = if(n=0,1,if(n=1,1,if(n=2,1,a(n-1)+2*sum{k=0..n-2, a(k)*a(n-1-k)})))
%F Recurrence: (n+1)*a(n) = 3*(2*n-1)*a(n-1) - (n-2)*a(n-2) - 8*(2*n-7)*a(n-3). - _Vaclav Kotesovec_, Nov 20 2012
%F G.f.: 1/4+(1-sqrt(16*x^3+x^2-6*x+1))/(4*x). - _Vaclav Kotesovec_, Nov 20 2012
%t CoefficientList[Series[1/4+(1-Sqrt[16*x^3+x^2-6*x+1])/(4*x),{x,0,20}],x] (* _Vaclav Kotesovec_, Nov 20 2012 *)
%Y Cf.: A025262, A056010, A006720.
%K easy,nonn
%O 0,4
%A _Paul Barry_, May 24 2009
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