A178074 Sequence with a (1,2) Somos-4 Hankel transform.

1, 0, -2, 1, 3, -5, 1, 10, -24, 18, 52, -182, 180, 348, -1474, 1733, 2407, -12557, 17145, 16561, -111041, 172625, 107783, -1006217, 1759149, 592699, -9265399, 18081483, 1574873, -86180045, 186977185, -25366798, -805909936, 1941467634, -657563052

Offset

0,3

Comments

Hankel transform is A178075(n+2).

Links

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

Formula

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)).

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

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

G.f.: 2 / (1 - x + 2*x^2 + sqrt(1 + 2*x + 5*x^2 + 4*x^4)). - Michael Somos, Aug 06 2014

Example

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 + ...

Mathematica

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 *)

Prog

(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
(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

Crossrefs

Cf. A178075.

Sequence in context: A058168 A367196 A058169 * A002249 A157127 A066748

Adjacent sequences: A178071 A178072 A178073 * A178075 A178076 A178077

Keyword

easy,sign

Author

Paul Barry, May 19 2010

Status

approved