A160703 Generalized Somos-4 Hankel determinant recurrence sequence.

1, 4, 20, 464, 17024, 1632256, 499253248, 211076251648, 432343353393152, 1184049383163822080, 10988005485374521475072, 249434575164299910905331712, 9434727599158183495094688022528

Offset: 0

Author

Paul Barry, May 24 2009

Keywords

• easy
• nonn

Comments

Hankel transform of A160702(n+1).

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..71
• Gouce Xin, Proof of the Somos-4 Hankel determinants conjecture, Advances in Applied Mathematics, Volume 42, Issue 2, February 2009, Pages 152-156.

Crossrefs

Cf. A006720.

Programs

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

Mathematica

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 *)

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 */

Formula

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) = a(n-3) * 2^(2*n - 3). a(-1) = a(0) = 1. - Michael Somos, Jun 15 2011