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A367391 Expansion of e.g.f. A(x) satisfying A(x)^2 = exp(x) * A(x*A(x)) with A(0) = 1.

%I A367391 #9 Jan 09 2024 08:50:43
%S A367391 1,1,3,28,569,19686,1015357,72213450,6732370465,794072741302,
%T A367391 115412704302581,20251767162061986,4220273910604275889,
%U A367391 1030325477950545779094,291316596476686970503693,94452315650030395608940066,34815037905775665043220138561,14478491178300336588521758911894
%N A367391 Expansion of e.g.f. A(x) satisfying A(x)^2 = exp(x) * A(x*A(x)) with A(0) = 1.
%C A367391 Note that if F(x)^2 = exp(x) * F(x*F(x)) with F(0) = 0, then F(x) is the e.g.f. of A367390.
%H A367391 Paul D. Hanna, <a href="/A367391/b367391.txt">Table of n, a(n) for n = 0..225</a>
%F A367391 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! and B(x) = x*A(x) satisfies the following formulas.
%F A367391 (1) A(x)^2 = exp(x) * A(x*A(x)).
%F A367391 Let B^n(x) denote the n-th iteration of B(x) = x*A(x), where B^(n+1)(x) = B( B^n(x) ) with B^0(x) = x, then
%F A367391 (2) log( A(x) ) = Sum_{n>=0} B^n(x) / 2^(n+1).
%F A367391 (3) B^n(x) = x*A(x)^(2^n - 1) / exp( Sum_{k=0..n-2} (2^(n-k-1) - 1) * B^k(x) ) for n > 1.
%F A367391 (3.a) B^2(x) = x*A(x)^3 / exp(x).
%F A367391 (3.b) B^3(x) = x*A(x)^7 / exp(3*x + B(x)).
%F A367391 (3.c) B^4(x) = x*A(x)^15 / exp(7*x + 3*B(x) + B^2(x)).
%F A367391 (3.d) B^5(x) = x*A(x)^31 / exp(15*x + 7*B(x) + 3*B^2(x) + B^3(x)).
%F A367391 (4) A( B^n(x) ) = A(x)^(2^n) / exp( Sum_{k=0..n-1} 2^(n-k-1) * B^k(x) ) for n > 0.
%F A367391 (4.a) A(B(x)) = A(x)^2 / exp(x).
%F A367391 (4.b) A(B^2(x)) = A(x)^4 / exp(2*x + B(x)).
%F A367391 (4.c) A(B^3(x)) = A(x)^8 / exp(4*x + 2*B(x) + B^2(x)).
%F A367391 (4.d) A(B^4(x)) = A(x)^16 / exp(8*x + 4*B(x) + 2*B^2(x) + B^3(x)).
%e A367391 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 28*x^3/3! + 569*x^4/4! + 19686*x^5/5! + 1015357*x^6/6! + 72213450*x^7/7! + 6732370465*x^8/8! + 794072741302*x^9/9! + 115412704302581*x^10/10! + ...
%e A367391 where A(x)^2 = exp(x) * A(x*A(x)) as can be seen from the following expansions
%e A367391 A(x)^2 = 1 + 2*x + 8*x^2/2! + 74*x^3/3! + 1416*x^4/4! + 46742*x^5/5! + 2333836*x^6/6! + 162237574*x^7/7! + ...
%e A367391 A(x*A(x)) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1161*x^4/4! + 40331*x^5/5! + 2073253*x^6/6! + 146835179*x^7/7! + ...
%e A367391 RELATED SERIES.
%e A367391 log(A(x)) = x + 2*x^2/2! + 21*x^3/3! + 460*x^4/4! + 16675*x^5/5! + 886926*x^6/6! + 64453095*x^7/7! + 6104710088*x^8/8! + 728774208459*x^9/9! + ...
%e A367391 Let B(x) = x*A(x), then log(A(x)) equals a sum over all iterations of B(x):
%e A367391 log(A(x)) = x/2 + B(x)/2^2 + B(B(x))/2^3 + B(B(B(x)))/2^4 + B(B(B(B(x))))/2^5 + ...
%e A367391 Successive iterations of B(x) = x*A(x) begin
%e A367391 B(x) = x + 2*x^2/2! + 9*x^3/3! + 112*x^4/4! + 2845*x^5/5! + 118116*x^6/6! + 7107499*x^7/7! + 577707600*x^8/8! + ...
%e A367391 B(B(x)) = x + 4*x^2/2! + 30*x^3/3! + 428*x^4/4! + 10760*x^5/5! + 430302*x^6/6! + 25021024*x^7/7! + ...
%e A367391 B(B(B(x))) = x + 6*x^2/2! + 63*x^3/3! + 1092*x^4/4! + 29625*x^5/5! + 1196658*x^6/6! + 68472705*x^7/7! + ...
%e A367391 B(B(B(B(x)))) = x + 8*x^2/2! + 108*x^3/3! + 2248*x^4/4! + 68200*x^5/5! + 2905524*x^6/6! + 168670432*x^7/7! + ...
%e A367391 etc.
%e A367391 where A(x) = exp(x/2 + B(x)/4 + B(B(x))/8 + B(B(B(x)))/16 + B(B(B(B(x))))/32 + ...).
%o A367391 (PARI) {a(n) = my(A=1, V=[1]); for(i=1,n, V = concat(V,0); A = Ser(V);
%o A367391 V[#V] = polcoeff( subst(A,x,x*A) - exp(-x +x*O(x^(#V)))*A^2, #V-1) ); n!*V[n+1]}
%o A367391 for(n=0,20, print1(a(n),", "))
%Y A367391 Cf. A367390.
%K A367391 nonn
%O A367391 0,3
%A A367391 _Paul D. Hanna_, Jan 08 2024