A261196 Expansion of sqrt(8*x + sqrt(1 + 64*x^2)).

1, 4, 8, -32, -160, 896, 5376, -33792, -219648, 1464320, 9957376, -68796416, -481574912, 3408068608, 24343347200, -175272099840, -1270722723840, 9268801044480, 67971207659520, -500840477491200, -3706219533434880, 27531916534087680, 205237923254108160

Offset

0,2

Comments

Signs are important to distinguish this from (for example) A098579.

Sqrt(A(x)) = 1 + 2*x + 2*x^2 - 20*x^3 - 42*x^4 + 572*x^5 ... defines another (new) integer sequence.

Links

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

Formula

G.f. A(x) satisfies: A(x)^4 = 1 + 16*x*A(x)^2.

A(x) = sqrt(G(4*x)), where G(x) is the g.f. of A182122.

A(x) * A(-x) = 1.

A(x) = sqrt(1 + 8*x + 32*x^2*C(-16*x^2)), where C(x) is the g.f. of A000108.

a(n) = A002420(n)*2^n*(-1)^(n*(n+1)/2). - M. F. Hasler, Aug 14 2015

Conjecture D-finite with recurrence: n*(n-1)*a(n) +(n-1)*(n-2)*a(n-1) +16*(2*n-3)*(2*n-5)*a(n-2) +16*(2*n-5)*(2*n-7)*a(n-3)=0. - R. J. Mathar, Jun 07 2016

Example

A(x) = 1 + 4*x + 8*x^2 - 32*x^3 - 160*x^4 + 896*x^5 + 5376*x^6 ...
A(x)^2 = 1 + 8*x + 32*x^2 + 0*x^3 - 512*x^4 + 0*x^5 +16384*x^6 ...
A(x)^4 = 1 + 16*x + 128*x^2 + 512*x^3 + 0*x^4 -8192*x^5 + 0*x^6 ...

Mathematica

CoefficientList[Series[Sqrt[8 x + Sqrt[1 + 64 x^2]], {x, 0, 45}], x] (* Vincenzo Librandi, Aug 12 2015 *)

Prog

(PARI) Vec(sqrt(8*x + sqrt(1 + 64*x^2))) \\ M. F. Hasler, Aug 14 2015
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt(8*x+Sqrt(1+64*x^2)))); // G. C. Greubel, Aug 12 2018

Crossrefs

Cf. A000108, A002420, A098579, A182122.

Sequence in context: A075398 A358408 A098579 * A222537 A208924 A032467

Adjacent sequences: A261193 A261194 A261195 * A261197 A261198 A261199

Keyword

sign,easy

Author

Werner Schulte, Aug 11 2015

Status

approved