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

1, 0, -1, -1, -1, -1, 1, 8, 23, 45, 55, -14, -317, -1095, -2459, -3574, -681, 16124, 64605, 159483, 260869, 134374, -906919, -4228769, -11317061, -20327731, -15742753, 52640154, 293447719, 847451759, 1648865921, 1636313816, -2986606297, -21074495982

Offset

0,8

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.

Formula

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

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. - Michael Somos, Jun 13 2010

G.f.: 2 / (1 + 2*x + x^2 + sqrt(1 - 4*x + 6*x^2 + x^4)). - Michael Somos, Jun 13 2010

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

Example

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 + ...

Mathematica

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 *)

Prog

(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 */
(PARI) 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
(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

Crossrefs

Cf. A006769.

Sequence in context: A114381 A139433 A226600 * A351966 A185257 A033951

Adjacent sequences: A178069 A178070 A178071 * A178073 A178074 A178075

Keyword

easy,sign

Author

Paul Barry, May 19 2010

Status

approved