OFFSET
1,3
COMMENTS
The true Euler characteristic alternate in sign. This sequence is the absolute value.
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies x = A - A^2/(1-A^2)^2.
G.f. A(x) satisfies [from Paul D. Hanna, Jul 14 2012]:
(1) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n)/(1-x^2)^(2*n)/n!.
(2) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)/(1-x^2)^(2*n)/n! ).
EXAMPLE
The number g(3)=2 is the Euler characteristic of relative homology of the punctured affine line A^1\{0} with respect to the divisor {1,2}. This is related to the cluster algebra of type A1.
G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 101*x^6 + 412*x^7 +...
where A(x - x^2/(1-x^2)^2) = x.
Related expansions [from Paul D. Hanna, Jul 14 2012]:
(1) A(x) = x + x^2/(1-x^2)^2 + d/dx x^4/(1-x^2)^4/2! + d^2/dx^2 x^6/(1-x^2)^6/3! + d^3/dx^3 x^8/(1-x^2)^8/4! +...
(2) log(A(x)/x) = x/(1-x^2)^2 + d/dx x^3/(1-x^2)^4/2! + d^2/dx^2 x^5/(1-x^2)^6/3! + d^3/dx^3 x^7/(1-x^2)^8/4! +...
MAPLE
eq:=x-G+G**2/(1-G**2)**2; solu:=solve(eq, G); taylor(solu, x, 16);
# second Maple program:
a:= n-> coeff(series(RootOf(x-A+A^2/(1-A^2)^2, A), x, n+1), x, n):
seq(a(n), n=1..30); # Alois P. Heinz, Feb 23 2017
MATHEMATICA
InverseSeries[x-x^2/(1-x^2)^2+x*O[x]^30, x] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Feb 23 2017, after Paul D. Hanna *)
PROG
(PARI) {a(n)=polcoeff(serreverse(x-x^2/(1-x^2)^2+x*O(x^n)), n)} \\ Paul D. Hanna, Jul 14 2012
for(n=1, 30, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^(2*m)/(1-x^2+x*O(x^n))^(2*m)/m!)); polcoeff(A, n)} \\ Paul D. Hanna, Jul 14 2012
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(2*m-1)/(1-x^2+x*O(x^n))^(2*m)/m!))); polcoeff(A, n)} \\ Paul D. Hanna, Jul 14 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
F. Chapoton, Dec 16 2009
STATUS
approved