cp's OEIS Frontend

A178072 Sequence with Hankel transform equal to the Somos-4 sequence A006769(n+2).

%I A178072 #35 May 11 2025 04:14:03
%S A178072 1,0,-1,-1,-1,-1,1,8,23,45,55,-14,-317,-1095,-2459,-3574,-681,16124,
%T A178072 64605,159483,260869,134374,-906919,-4228769,-11317061,-20327731,
%U A178072 -15742753,52640154,293447719,847451759,1648865921,1636313816,-2986606297,-21074495982
%N A178072 Sequence with Hankel transform equal to the Somos-4 sequence A006769(n+2).
%H A178072 G. C. Greubel, <a href="/A178072/b178072.txt">Table of n, a(n) for n = 0..1000</a>
%H A178072 Paul Barry, <a href="https://arxiv.org/abs/2306.05025">Integer sequences from elliptic curves</a>, arXiv:2306.05025 [math.NT], 2023.
%F A178072 a(n) = Sum_{k=0..floor(n/2)} (C(n-k,k)/(n-2*k+1))*Sum_{i=0..min(k,n-2*k)} C(k,i)*C(n-k-i-1,n-2*k-i)*2^(n-2*k-i)*(-1)^(k-i).
%F A178072 From _Michael Somos_, Jun 13 2010: (Start)
%F A178072 G.f. A(x) satisfies 0 = f(x, A(x)) where f(x, y) = (2*x + x^3) * y^2 - (1 + x)^2 * y + 1.
%F A178072 G.f.: 2 / (1 + 2*x + x^2 + sqrt(1 - 4*x + 6*x^2 + x^4)). (End)
%F A178072 Conjecture: 2*n*(n+1)*a(n) +2*n*(n+1)*a(n-1) -3*(3*n-1)*(3*n-4)*a(n-2) +(61*n^2-191*n+36)*a(n-3) +6*(-2*n^2+2*n-1)*a(n-4) +2*(5*n-1)*(4*n-15)*a(n-5) +n*(n-5)*a(n-6) +(5*n-1)*(n-6)*a(n-7)=0. - _R. J. Mathar_, Jun 14 2016
%e A178072 G.f. = 1 - x^2 - x^3 - x^4 - x^5 + x^6 + 8*x^7 + 23*x^8 + 45*x^9 + 55*x^10 + ...
%t A178072 CoefficientList[Series[2/(1+2*x+x^2+Sqrt[1-4*x+6*x^2+x^4]), {x, 0, 40}], x] (* _G. C. Greubel_, Sep 22 2018 *)
%o A178072 (PARI) {a(n) = if( n<0, 0, polcoeff( 2 / (1 + 2*x + x^2 + sqrt(1 - 4*x + 6*x^2 + x^4 + x*O(x^n))), n))}; /* _Michael Somos_, Jun 13 2010 */
%o A178072 (PARI) my(x='x+O('x^50)); Vec(2/(1+2*x+x^2+(1-4*x+6*x^2+x^4)^(1/2))) \\ _Altug Alkan_, Sep 23 2018
%o A178072 (Magma) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(2/(1+2*x+x^2+Sqrt(1-4*x+6*x^2+x^4)))); // _G. C. Greubel_, Sep 22 2018
%Y A178072 Cf. A006769.
%K A178072 easy,sign
%O A178072 0,8
%A A178072 _Paul Barry_, May 19 2010