A143045 - OEIS

%I #18 Dec 27 2013 20:26:46

%S 1,1,-2,-3,10,18,-68,-131,530,1062,-4476,-9198,39844,83332,-368136,

%T -780003,3497058,7483806,-33940940,-73210874,335103340,727473084,

%U -3355045304,-7322240718,33982884884,74498594492,-347600543192,-764936992764,3585459509640,7916276980872,-37253166379536

%N G.f.: A(x) = x + A(-x)^2.

%H Paul D. Hanna, <a href="/A143045/b143045.txt">Table of n, a(n) for n = 1..100</a>

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

%F G.f. satisfies: A(x) = x + ( x - A(x)^2 )^2.

%F G.f. satisfies: [A(x)^3 - A(-x)^3]/2 = x*[A(x) + A(-x)]/2.

%F Recurrence: 27*(n-2)*(n-1)*n*(16*n^3 - 160*n^2 + 521*n - 552)*a(n) = 18*(n-2)*(n-1)*(32*n - 105)*a(n-1) - 12*(n-2)*(384*n^5 - 5376*n^4 + 29288*n^3 - 77560*n^2 + 99709*n - 49665)*a(n-2) + 48*(2*n - 7)*(32*n^3 - 248*n^2 + 623*n - 505)*a(n-3) + 64*(n-4)*(2*n - 9)*(2*n - 7)*(16*n^3 - 112*n^2 + 249*n - 175)*a(n-4). - _Vaclav Kotesovec_, Dec 27 2013

%F Limit n->infinity |a(n)|^(1/n) = 4/3*sqrt(3+2*sqrt(3)) = 3.3899463424498833... - _Vaclav Kotesovec_, Dec 27 2013

%e A(x) = x + x^2 - 2*x^3 - 3*x^4 + 10*x^5 + 18*x^6 - 68*x^7 - 131*x^8 +...

%e A(x)^2 = x^2 + 2*x^3 - 3*x^4 - 10*x^5 + 18*x^6 + 68*x^7 - 131*x^8 - 530*x^9 +...

%e A(x)^3 = x^3 + 3*x^4 - 3*x^5 - 20*x^6 + 18*x^7 + 153*x^8 - 131*x^9 -++-...

%e Note that a bisection of A^3 equals a bisection of A.

%t Rest[CoefficientList[1 + InverseSeries[Series[(Sqrt[1+4*x-4*x^2]-1)/2 + x^2, {x, 0, 20}], x],x]] (* _Vaclav Kotesovec_, Dec 27 2013 *)

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

%o (PARI) {a(n)=local(A=x+x^2);for(i=0,n,A=x+subst(A,x,(-x+x*O(x^n)))^2);polcoeff(A,n)}

%o (PARI) {a(n)=local(A=x+x*O(x^n));for(i=0,n,A = x + x^2 - 2*x*A^2 + A^4);polcoeff(A,n)}

%Y Cf. A143046, A143048, A143049, A213252, A100238 (related by series reversion).

%K sign

%O 1,3

%A _Paul D. Hanna_, Jul 19 2008, Jul 20 2008