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%I #2 Mar 30 2012 18:37:16
%S 1,1,3,63,6732,3414312,10221878106,243813944182248,
%T 50538758405328815616,87376772859536771916909012,
%U 1235009698863206337006094872463887,142641072494398006081741872595533545306244
%N G.f. A(x) satisfies: A(x) = 1 + x*Sum_{n>=0} log( A(3^n*x) )^n / n!.
%F G.f. A(x) satisfies: a(n+1) = [x^n] A(x)^(3^n) for n>=0, with a(0)=1.
%e G.f.: A(x) = 1 + x + 3*x^2 + 63*x^3 + 6732*x^4 + 3414312*x^5 +...
%e SERIES REPRESENTATION:
%e A(x) = 1 + x*[1 + log(A(3x)) + log(A(9x))^2/2! + log(A(27x))^3/3! +...+ log(A(3^n*x))^n/n! +...].
%e ...
%e GENERATED BY POWERS OF G.F.:
%e a(n+1) equals the coefficient of x^n in A(x)^(3^n) for n>=0;
%e the coefficients of A(x)^(3^n) begin:
%e A^(3^0): [(1), 1, 3, 63, 6732, 3414312, 10221878106, ...];
%e A^(3^1): [1, (3), 12, 208, 20610, 10284678, 30686274630, ...];
%e A^(3^2): [1, 9, (63), 867, 66330, 31246902, 92246164932, ...];
%e A^(3^3): [1, 27, 432, (6732), 273024, 97968096, 278472473082, ...];
%e A^(3^4): [1, 81, 3483, 109863, (3414312), 385422948, 853280745822, ...];
%e A^(3^5): [1, 243, 30132, 2553768, 168586110, (10221878106), ...];
%e In the above table, the diagonal forms this sequence shift left.
%o (PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, Vec(Ser(A)^(3^(#A-1)))[ #A])); A[n+1]}
%Y Cf. A132695, A156905.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 04 2009
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