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A200402 G.f. satisfies: A(x) = exp( Sum_{n>=1} A(-x^n)^3 * x^n/n ).

%I A200402 #22 Mar 24 2017 03:50:00
%S A200402 1,1,-2,-5,24,81,-439,-1590,9144,34451,-206641,-799196,4936378,
%T A200402 19442800,-122613798,-489411508,3134773097,12640278932,-81948641010,
%U A200402 -333099985517,2180523864984,8920922434686,-58861487584914,-242105281357185,1608002839956522,6643707274089977,-44372373955131024
%N A200402 G.f. satisfies: A(x) = exp( Sum_{n>=1} A(-x^n)^3 * x^n/n ).
%C A200402 Compare g.f. to the trivial identity: G(x) = exp(Sum_{n>=1} G(-x^n)*x^n/n) where G(x) = 1+x.
%H A200402 Alois P. Heinz, <a href="/A200402/b200402.txt">Table of n, a(n) for n = 0..1347</a>
%F A200402 Equals the Euler transformation of the coefficients in A(-x)^3, where A(x) is the g.f. of this sequence.
%e A200402 G.f.: A(x) = 1 + x - 2*x^2 - 5*x^3 + 24*x^4 + 81*x^5 - 439*x^6 +...
%e A200402 where
%e A200402 log(A(x)) = A(-x)^3*x + A(-x^2)^3*x^2/2 + A(-x^3)^3*x^3/3 + A(-x^4)^3*x^4/4 +...
%e A200402 The coefficients in A(-x)^3 begin:
%e A200402 [1,-3,-3,26,48,-444,-920,9126,19587,-204214,-449496,4841001,...]
%e A200402 and the g.f. may be expressed by the Euler product:
%e A200402 A(x) = 1/((1-x)^1*(1-x^2)^-3*(1-x^3)^-3*(1-x^4)^26*(1-x^5)^48*(1-x^6)^-444*(1-x^7)^-920*(1-x^8)^9126*...).
%p A200402 b:= proc(n) option remember; add(a(i)*a(n-i), i=0..n) end:
%p A200402 g:= proc(n) option remember; (-1)^n*add(b(i)*a(n-i), i=0..n) end:
%p A200402 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A200402       d*g(d-1), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p A200402     end:
%p A200402 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jan 24 2017
%t A200402 b[n_] := b[n] = Sum[a[i]*a[n-i], {i, 0, n}];
%t A200402 g[n_] := g[n] = (-1)^n*Sum[b[i]*a[n-i], {i, 0, n}];
%t A200402 a[n_] := a[n] = If[n==0, 1, Sum[DivisorSum[j, #*g[#-1]&]*a[n-j], {j, 1, n}]/n];
%t A200402 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 24 2017, after _Alois P. Heinz_ *)
%o A200402 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,subst(A^3,x,-x^m)*x^m/m)+x*O(x^n)));polcoeff(A,n)}
%Y A200402 Cf. A052751, A200438.
%K A200402 sign
%O A200402 0,3
%A A200402 _Paul D. Hanna_, Nov 17 2011