A143045 G.f.: A(x) = x + A(-x)^2.

1, 1, -2, -3, 10, 18, -68, -131, 530, 1062, -4476, -9198, 39844, 83332, -368136, -780003, 3497058, 7483806, -33940940, -73210874, 335103340, 727473084, -3355045304, -7322240718, 33982884884, 74498594492, -347600543192, -764936992764, 3585459509640, 7916276980872, -37253166379536

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..100

Formula

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

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

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

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

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

Example

A(x) = x + x^2 - 2*x^3 - 3*x^4 + 10*x^5 + 18*x^6 - 68*x^7 - 131*x^8 +...
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 +...
A(x)^3 = x^3 + 3*x^4 - 3*x^5 - 20*x^6 + 18*x^7 + 153*x^8 - 131*x^9 -++-...
Note that a bisection of A^3 equals a bisection of A.

Mathematica

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 *)

Prog

(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))}
(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)}
(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)}

Crossrefs

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

Sequence in context: A057507 A233895 A163467 * A156909 A215121 A329850

Adjacent sequences: A143042 A143043 A143044 * A143046 A143047 A143048

Keyword

sign

Author

Paul D. Hanna, Jul 19 2008, Jul 20 2008

Status

approved