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%I #2 Mar 30 2012 18:37:21
%S 1,1,4,43,1288,111874,28513474,21447796411,47864888436352,
%T 318138497623694374,6313455845922307358476,
%U 374710369870861989860068222,66586966437639051221223067992430
%N G.f.: Sum_{n>=0} a(n)*x^n/3^(n(n-1)/2) = exp( Sum_{n>=1} A(x/3^n)^n*x^n/n ).
%C More generally, coefficients a(n,q) that satisfy:
%C . A(x,q) = Sum_{n>=0} a(n,q)*x^n/q^(n(n-1)/2) such that
%C . A(x,q) = exp( Sum_{n>=1} A(x/q^n,q)^n * x^n/n )
%C begin:
%C a(0,q) = 1; a(1,q) = 1; a(2,q) = q + 1; a(3,q) = q^3 + q^2 + 2*q + 1;
%C a(4,q) = q^6 + q^5 + 5/2*q^4 + 3*q^3 + 5/2*q^2 + 3*q + 1;
%C a(5,q) = q^10 + q^9 + 5/2*q^8 + 5*q^7 + 11/2*q^6 + 4*q^5 + 15/2*q^4 + 6*q^3 + 9/2*q^2 + 4*q + 1; ...
%C Conjecture: {a(n,q), n>=0} are integers when q is a positive integer.
%C Conjecture: a(n,q) = 1 (mod q) for n>=0 when q is a positive integer. [From _Paul D. Hanna_, Apr 29 2010]
%e G.f.: A(x) = 1 + x + 4*x^2/3 + 43*x^3/3^3 + 1288*x^4/3^6 + 111874*x^5/3^10 +...
%e log(A(x)) = A(x/3)*x + A(x/9)^2*x^2/2 + A(x/27)^3*x^3/3 + A(x/81)^4*x^4/4 +...
%o (PARI) {a(n)=local(A=1+x); for(n=2, n, A=exp(sum(k=1, n, subst(A, x, x/3^k+x*O(x^n))^k*x^k/k))); 3^(n*(n-1)/2)*polcoeff(A, n)}
%Y Cf. A159592.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 27 2010
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