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A163966 G.f. satisfies: the coefficient of x^n in A(x)^n = 3^(n^2) for n>=0.

%I A163966 #2 Mar 30 2012 18:37:18
%S A163966 1,3,36,6336,10701720,169328019456,25013229623639712,
%T A163966 34185168163500076044288,429209759636992944567490013280,
%U A163966 49269599503729281688381600977015275520
%N A163966 G.f. satisfies: the coefficient of x^n in A(x)^n = 3^(n^2) for n>=0.
%F A163966 G.f. satisfies: A(x) = exp( Sum_{n>=1} 3^(n^2)*[x/A(x)]^n/n ).
%F A163966 Let G(x) = exp(Sum_{n>=1} 3^(n^2)*x^n/n) = g.f. of A155203, then:
%F A163966 (1) A(x) = G(x/A(x)) and A(x*G(x)) = G(x) ;
%F A163966 (2) A(x) = x/Series_Reversion[x*G(x)] ;
%F A163966 (3) [x^n] A(x)^(n+1)/(n+1) = [x^n] G(x) = A155203(n) ;
%F A163966 (4) [x^n] A(x)^(n+m)*m/(n+m) = [x^n] G(x)^m for |n+m|>0, n>=0.
%e A163966 G.f.: A(x) = 1 + 3*x + 36*x^2 + 6336*x^3 + 10701720*x^4 +...
%e A163966 The coefficients in the successive powers of A(x) begin:
%e A163966 [1,(3), 36, 6336, 10701720, 169328019456, 25013229623639712,...];
%e A163966 [1, 6,(81), 12888, 21442752, 338720705424, 50027476026064896,...];
%e A163966 [1, 9, 135,(19683), 32224068, 508178240640, 75042739500374376,...];
%e A163966 [1, 12, 198, 26748,(43046721), 677700811728, 100059020340421248,...];
%e A163966 [1, 15, 270, 34110, 53911845,(847288609443), 125076318840827460,...];
%e A163966 [1, 18, 351, 41796, 64820655, 1016941828914,(150094635296999121),...];
%e A163966 [1, 21, 441, 49833, 75774447, 1186660669887, 175113970005142539,...];
%e A163966 [1, 24, 540, 58248, 86774598, 1356445336968, 200134323262280988,...];
%e A163966 ...
%e A163966 The above terms in parenthesis = [x^n] A(x)^n = 3^(n^2) for n=1,2,3,...
%e A163966 The main diagonal = [x^n] A(x)^(n+1) = (n+1)*A155203(n):
%e A163966 [1, 2*3, 3*45, 4*6687, 5*10782369, 6*169490304819, ...].
%o A163966 (PARI) {a(n)=local(G=exp(sum(m=1, n, 3^(m^2)*x^m/m)+x*O(x^n))); polcoeff(x/serreverse(x*G), n)}
%o A163966 (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, 3^(m^2)*(x/A)^m/m))); polcoeff(A, n)}
%Y A163966 Cf. A164764, A155203.
%K A163966 nonn
%O A163966 0,2
%A A163966 _Paul D. Hanna_, Aug 28 2009