A182492 - OEIS

%I #18 Sep 08 2022 08:45:55

%S 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,

%T 0,-429,0,0,0,1430,0,0,0,-4862,0,0,0,16796,0,0,0,-58786,0,0,0,208012,

%U 0,0,0,-742900,0,0,0,2674440,0,0,0,-9694845,0,0,0

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

%C 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.

%H G. C. Greubel, <a href="/A182492/b182492.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%F 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))))))))).

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

%F 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

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

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

%o (PARI) {a(n) = if( n<0, 0, polcoeff( 1 - x - (1 - sqrt(1 + 4*x^4 + x^2 * O(x^n))) / (2*x), n))}

%o (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))}

%o (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))}

%o (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

%Y Cf. A000108, A087960.

%K sign

%O 0,12

%A _Michael Somos_, May 02 2012