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%I #16 Mar 24 2017 04:41:41
%S 1,1,-1,-2,5,14,-40,-119,351,1083,-3291,-10424,32562,105066,-334666,
%T -1094595,3536043,11686231,-38172425,-127199414,419230644,1406346735,
%U -4669311299,-15750517780,52616257231,178312867791,-598779740235,-2037290707630,6871904761413,23461177498832
%N G.f. satisfies: A(x) = exp( Sum_{n>=1} A(-x^n)^2 * x^n/n ).
%C Compare g.f. to the trivial identity: G(x) = exp(Sum_{n>=1} G(-x^n)*x^n/n) where G(x) = 1+x.
%C abs(a(n+1)/a(n)) tends to 3.576353722518567708610064857260994390208457341780918501933217195112489... . - _Vaclav Kotesovec_, Mar 24 2017
%H Alois P. Heinz, <a href="/A200438/b200438.txt">Table of n, a(n) for n = 0..1816</a>
%F Equals the Euler transformation of the coefficients in A(-x)^2, where A(x) is the g.f. of this sequence.
%e G.f.: A(x) = 1 + x - x^2 - 2*x^3 + 5*x^4 + 14*x^5 - 40*x^6 - 119*x^7 +...
%e where
%e 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 +...
%e The coefficients in A(-x)^2 begin:
%e [1,-2,-1,6,7,-42,-58,366,513,-3406,-4846,33310,48304,-339446,...]
%e and the g.f. may be expressed by the Euler product:
%e 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*...).
%p b:= proc(n) option remember; (-1)^n*add(a(i)*a(n-i), i=0..n) end:
%p a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p d*b(d-1), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jan 24 2017
%t 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 *)
%o (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)}
%Y Cf. A200402.
%K sign
%O 0,4
%A _Paul D. Hanna_, Nov 17 2011
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