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A160703 Generalized Somos-4 Hankel determinant recurrence sequence.

%I A160703 #17 Sep 08 2022 08:45:45
%S A160703 1,4,20,464,17024,1632256,499253248,211076251648,432343353393152,
%T A160703 1184049383163822080,10988005485374521475072,
%U A160703 249434575164299910905331712,9434727599158183495094688022528
%N A160703 Generalized Somos-4 Hankel determinant recurrence sequence.
%C A160703 Hankel transform of A160702(n+1).
%C A160703 In general, we can conjecture that the Hankel transform a(n) of f(n+1), where f(n)=if(n=0,1,if(n=1,1,if(n=2,1,r*f(n-1)+s*sum{k=0..n-2, f(k)*f(n-1-k)}))) satisfies the generalized Somos-4 recurrence: a(n)=(s^2*a(n-1)*a(n-3)+s^3*(2*s+r-2)*a(n-2)^2)/a(n-4). The case r=s=1 is proved in the Xin reference.
%H A160703 G. C. Greubel, <a href="/A160703/b160703.txt">Table of n, a(n) for n = 0..71</a>
%H A160703 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 A160703 a(n) = (4*a(n-1)*a(n-3)+24*a(n-2)^2)/a(n-4), a(0)=1, a(1)=4, a(2)=20, a(3)=464.
%F A160703 a(-n) = a(n-3) * 2^(2*n - 3). a(-1) = a(0) = 1. - _Michael Somos_, Jun 15 2011
%t A160703 RecurrenceTable[{a[n] == (4*a[n-1]*a[n-3] + 24*a[n-2]^2)/a[n-4], a[0] == 1, a[1] == 4, a[2] == 20, a[3] == 464}, a, {n, 0, 20}] (* _G. C. Greubel_, Sep 21 2018 *)
%o A160703 (PARI) {a(n) = if( n<-1, a( -n - 3) * 2^( -2*n - 3), if( n<3, [1, 1, 4, 20][n + 2], (4 * a(n-1) * a(n-3) + 24 * a(n-2)^2) / a(n-4)))} /* _Michael Somos_, Jun 15 2011 */
%o A160703 (Magma) I:=[1,4,20,464]; [n le 4 select I[n] else (4*Self(n-1)*Self(n-3) +24*Self(n-2)^2)/Self(n-4): n in [1..30]]; // _G. C. Greubel_, Sep 21 2018
%Y A160703 Cf. A006720.
%K A160703 easy,nonn
%O A160703 0,2
%A A160703 _Paul Barry_, May 24 2009