A182492 Expansion of 1 - x - (1 - sqrt(1 + 4*x^4)) / (2*x) in powers of x.

1, -1, 0, 1, 0, 0, 0, -1, 0, 0, 0, 2, 0, 0, 0, -5, 0, 0, 0, 14, 0, 0, 0, -42, 0, 0, 0, 132, 0, 0, 0, -429, 0, 0, 0, 1430, 0, 0, 0, -4862, 0, 0, 0, 16796, 0, 0, 0, -58786, 0, 0, 0, 208012, 0, 0, 0, -742900, 0, 0, 0, 2674440, 0, 0, 0, -9694845, 0, 0, 0

Offset

0,12

Comments

HANKEL transform of sequence is the period 4 sequence [ 1, -1, -1, 1, ...] A087960 and the HANKEL transform of sequence omitting a(0) is the period 4 sequence [ -1, -1, -1, 1, ...]. This is the unique sequence with that property.

Links

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

Formula

G.f.: 1 - x - (1 - sqrt(1 + 4*x^4)) / (2*x).

G.f. A(x) satisfies A(x) = 1 - x / (1 - 2*x + 2*x^2) * A(x)^2.

G.f. A(x) satisfies A(x) = 1 - x + x^3 - x^7 / (1 - x + x^2 + x^4 + x * A(x)) = 1 / (1 + x / (1 - x / (1 + x / (1 - x / (1 - x / (1 + x / (1 - x / (1 + x * A(x))))))))).

a(4*n) = a(4*n + 1) = 0 unless n=0. a(4*n + 2) = 0. a(4*n + 3) = (-1)^n * A000108(n).

D-finite with recurrence: n*(n+1)*a(n) +n*(n+1)*a(n-1) +(n+2)*(n-1)*a(n-2) +(n+3)*(n-2)*a(n-3) +4*n*(n-5)*a(n-4) +4*(n+1)*(n-6)*a(n-5) +4*(n+2)*(n-7)*a(n-6) +4*(n+3)*(n-8)*a(n-7)=0. - R. J. Mathar, Jun 08 2016

Example

G.f. = 1 - x + x^3 - x^7 + 2*x^11 - 5*x^15 + 14*x^19 - 42*x^23 + 132*x^27 + ...

Mathematica

CoefficientList[Series[1-x -(1-Sqrt[1+4*x^4])/(2*x), {x, 0, 50}], x] (* G. C. Greubel, Aug 11 2018 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( 1 - x - (1 - sqrt(1 + 4*x^4 + x^2 * O(x^n))) / (2*x), n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = 1 - x / (1 - 2*x + 2*x^2) * A^2); polcoeff( A, n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = 1 + O(x); for( k=1, ceil(n / 8), A = 1 - x + x^3 - x^7 / (1 - x + x^2 + x^4 + x*A)); polcoeff( A, n))}
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(1-x -(1-Sqrt(1+4*x^4))/(2*x))); // G. C. Greubel, Aug 11 2018

Crossrefs

Cf. A000108, A087960.

Sequence in context: A283666 A131427 A153198 * A222898 A113044 A333792

Adjacent sequences: A182489 A182490 A182491 * A182493 A182494 A182495

Keyword

sign

Author

Michael Somos, May 02 2012

Status

approved