OFFSET
0,4
COMMENTS
Compare g.f. to the trivial identity: G(x) = exp(Sum_{n>=1} G(-x^n)*x^n/n) where G(x) = 1+x.
abs(a(n+1)/a(n)) tends to 3.576353722518567708610064857260994390208457341780918501933217195112489... . - Vaclav Kotesovec, Mar 24 2017
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1816
FORMULA
Equals the Euler transformation of the coefficients in A(-x)^2, where A(x) is the g.f. of this sequence.
EXAMPLE
G.f.: A(x) = 1 + x - x^2 - 2*x^3 + 5*x^4 + 14*x^5 - 40*x^6 - 119*x^7 +...
where
log(A(x)) = A(-x)^2*x + A(-x^2)^2*x^2/2 + A(-x^3)^2*x^3/3 + A(-x^4)^2*x^4/4 +...
The coefficients in A(-x)^2 begin:
[1,-2,-1,6,7,-42,-58,366,513,-3406,-4846,33310,48304,-339446,...]
and the g.f. may be expressed by the Euler product:
A(x) = 1/((1-x)^1*(1-x^2)^-2*(1-x^3)^-1*(1-x^4)^6*(1-x^5)^7*(1-x^6)^-42*(1-x^7)^-58*(1-x^8)^366*...).
MAPLE
b:= proc(n) option remember; (-1)^n*add(a(i)*a(n-i), i=0..n) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
d*b(d-1), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jan 24 2017
MATHEMATICA
A200438List[n_] := Module[{A, x, i}, A = 1+x; For[i=1, i <= n, i++, A = Exp[Sum[(A^2 /. x -> -x^m)*x^m/m, {m, 1, n}] + x*O[x]^n // Normal]]; CoefficientList[A + O[x]^n, x]]; A200438List[30] (* Jean-François Alcover, Mar 24 2017, adapted from PARI *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, subst(A^2, x, -x^m)*x^m/m)+x*O(x^n))); polcoeff(A, n)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Nov 17 2011
STATUS
approved