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A366227 O.g.f. A(x) satisfies: A(x) = 1 + x*Sum_{n>=0} 3^n * log( A(2^n*x) )^n / n!.

%I A366227 #5 Oct 17 2023 11:28:10
%S A366227 1,1,6,138,8648,1272948,424058592,334836466656,728593565874816,
%T A366227 5632989888855720864,184539760855097635059200,
%U A366227 25027477244647424010315231744,13206715998089387470949589465286656,26431031766456352400292737393044784872448,199091399877503863934385670788355318673030504448
%N A366227 O.g.f. A(x) satisfies: A(x) = 1 + x*Sum_{n>=0} 3^n * log( A(2^n*x) )^n / n!.
%C A366227 In general, we have the following identity:
%C A366227 given A(x) = Sum_{n>=0} a(n)*x^n satisfies
%C A366227 A(x) = 1 + x*Sum_{n>=0} p^n * log( A(q^n*x) )^n / n!,
%C A366227 then a(n+1) = [x^n] A(x)^(p*q^n) for n >= 0, with a(0)=1,
%C A366227 for arbitrary fixed parameters p and q.
%C A366227 Here, p = 3 and q = 2.
%F A366227 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A366227 (1) A(x) = 1 + x*Sum_{n>=0} 3^n * log( A(2^n*x) )^n / n!.
%F A366227 (2) a(n+1) = [x^n] A(x)^(3*2^n) for n >= 0, with a(0)=1.
%e A366227 G.f.: A(x) = 1 + x + 6*x^2 + 138*x^3 + 8648*x^4 + 1272948*x^5 + 424058592*x^6 + 334836466656*x^7 + 728593565874816*x^8 + ...
%e A366227 where
%e A366227 A(x) = 1 + x*[1 + 3*log(A(2*x)) + 3^2*log(A(2^2*x))^2/2! + 3^3*log(A(2^2*x))^3/3! + ... + 3^n*log(A(2^n*x))^n/n! + ...].
%e A366227 RELATED SERIES.
%e A366227 log(A(x)) = x + 11*x^2/2 + 397*x^3/3 + 33991*x^4/4 + 6318201*x^5/5 + 2536406543*x^6/6 + 2340834765809*x^7/7 + ...
%e A366227 RELATED TABLE.
%e A366227 The table of coefficients of x^k in A(x)^(3*2^n) begins:
%e A366227 n=0: [1,   3,    21,     451,    26898,    3876222, ...];
%e A366227 n=1: [1,   6,    51,    1028,    56943,    7932774, ...];
%e A366227 n=2: [1,  12,   138,    2668,   128823,   16653720, ...];
%e A366227 n=3: [1,  24,   420,    8648,   340722,   37135560, ...];
%e A366227 n=4: [1,  48,  1416,   37456,  1272948,   97890096, ...];
%e A366227 n=5: [1,  96,  5136,  210848,  8146728,  424058592, ...];
%e A366227 n=6: [1, 192, 19488, 1407808, 83154768, 4578119616, 334836466656, ...]; ...
%e A366227 in which the main diagonal equals this sequence shift left,
%e A366227 illustrating that a(n+1) = [x^n] A(x)^(3*2^n) for n >= 0.
%o A366227 (PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, Vec(Ser(A)^(3*2^(#A-1)))[ #A])); A[n+1]}
%o A366227 for(n=0, 15, print1(a(n), ", "))
%o A366227 (PARI) {a(n) = my(A=1+x); for(i=1, n, A = 1 + x*sum(m=0, #A, 3^m*log( subst(Ser(A), x, 2^m*x +x*O(x^n)))^m/m!) ); polcoeff(A, n)}
%o A366227 for(n=0, 15, print1(a(n), ", "))
%Y A366227 Cf. A132695, A156904, A156905, A181444, A366226.
%K A366227 nonn
%O A366227 0,3
%A A366227 _Paul D. Hanna_, Oct 17 2023