A200438 G.f. satisfies: A(x) = exp( Sum_{n>=1} A(-x^n)^2 * x^n/n ).

1, 1, -1, -2, 5, 14, -40, -119, 351, 1083, -3291, -10424, 32562, 105066, -334666, -1094595, 3536043, 11686231, -38172425, -127199414, 419230644, 1406346735, -4669311299, -15750517780, 52616257231, 178312867791, -598779740235, -2037290707630, 6871904761413, 23461177498832

Offset: 0

Author

Paul D. Hanna, Nov 17 2011

Keywords

• sign

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

Examples

			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*...).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1816

Crossrefs

Cf. A200402.

Programs

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 *)

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)}

Formula

Equals the Euler transformation of the coefficients in A(-x)^2, where A(x) is the g.f. of this sequence.