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Sequence with a (1,2) Somos-4 Hankel transform.
2

%I #8 Sep 08 2022 08:45:53

%S 1,0,-2,1,3,-5,1,10,-24,18,52,-182,180,348,-1474,1733,2407,-12557,

%T 17145,16561,-111041,172625,107783,-1006217,1759149,592699,-9265399,

%U 18081483,1574873,-86180045,186977185,-25366798,-805909936,1941467634,-657563052

%N Sequence with a (1,2) Somos-4 Hankel transform.

%C Hankel transform is A178075(n+2).

%H G. C. Greubel, <a href="/A178074/b178074.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..floor(n/2)} ((C(n-k,k)/(n-2k+1))*Sum_{i=0..k}(C(k,i)*C(n-k-i-1,n-2*k-i)*(-1)^(n-2*k-i)*(-1)^i*(-1)^(k-i)).

%F G.f. A(x) satisfies 0 = f(x, A(x)) where f(x, y) = (x + x^3) * y^2 + (1 - x + 2*x^2) * y - 1. - _Michael Somos_, Aug 06 2014

%F 0 = a(n)*(4*n + 4) + a(n+2) * (9*n + 24) + a(n+3)*(2*n + 11) + a(n+4)*(6*n + 27) + a(n+5)*(2*n + 11) + a(n+6)*(n+7) if n>-2. - _Michael Somos_, Aug 06 2014

%F G.f.: 2 / (1 - x + 2*x^2 + sqrt(1 + 2*x + 5*x^2 + 4*x^4)). - _Michael Somos_, Aug 06 2014

%e G.f. = 1 - 2*x^2 + x^3 + 3*x^4 - 5*x^5 + x^6 + 10*x^7 - 24*x^8 + 18*x^9 + ...

%t CoefficientList[Series[2/(1 -x +2*x^2 +Sqrt[1 +2*x +5*x^2 +4*x^4]), {x, 0, 50}], x] (* _G. C. Greubel_, Sep 22 2018 *)

%o (PARI) x='x+O('x^50); Vec(2/(1-x+2*x^2+sqrt(1+2*x+5*x^2+4*x^4))) \\ _G. C. Greubel_, Sep 22 2018

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(2/(1-x+2*x^2+Sqrt(1+2*x+5*x^2+4*x^4)))); // _G. C. Greubel_, Sep 22 2018

%Y Cf. A178075.

%K easy,sign

%O 0,3

%A _Paul Barry_, May 19 2010