OFFSET
1,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..800
FORMULA
G.f.: Series_Reversion( x+x^2 + x^2/(1-x-x^2) ).
G.f. A(x) satisfies: A(x)^4 + 2*A(x)^3 - (1+x)*A(x)^2 - (1+x)*A(x) + x = 0.
a(n) ~ (-1)^(n+1) * sqrt((1 + s - s^2)/(1 - r + 6*s - 6*s^2)) / (2*sqrt(Pi) * n^(3/2) * r^(n-1/2)), where r = 0.13635433405134757733687605093606926... and s = 0.27937563184993776460598554971985830... are roots of the system of equations 2*s^2*(3-2*s) = (1-r)*(1-2*s), r + s^3*(2-s) = (1-r)*s*(1-s). - Vaclav Kotesovec, Mar 06 2016
EXAMPLE
G.f.: A(x) = x - 2*x^2 + 7*x^3 - 32*x^4 + 164*x^5 - 901*x^6 + 5188*x^7 - 30898*x^8 + 188762*x^9 - 1176352*x^10 +...
where
A(x) = (1+x)*A(x)^2/x + (1+x^2)*A(x)^4/x^2 + (1+x^3)*A(x)^6/x^3 + (1+x^4)*A(x)^8/x^4 + (1+x^5)*A(x)^10/x^5 + (1+x^6)*A(x)^12/x^6 +...
RELATED SERIES.
Let G(x) denote the series reversion of the g.f. A(x), then
G(x) = x + 2*x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 8*x^7 + 13*x^8 + 21*x^9 + 34*x^10 + 55*x^11 + 89*x^12 +...+ Fibonacci(n-1)*x^n +...
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[x+x^2 + x^2/(1-x-x^2), {x, 0, 40}], x], x]] (* Vaclav Kotesovec, Mar 06 2016 *)
PROG
(PARI) {a(n) = my(A=x); for(i=1, n, A = A^2/sum(m=1, n, (1 + x^m)/x^m * (A +x*O(x^n))^(2*m) ) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = polcoeff( serreverse( x+x^2 + x^2/(1-x-x^2 +x*O(x^n)) ), n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Feb 24 2016
STATUS
approved