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E.g.f. A(x) satisfies: A(x) = 1 + Integral Sum_{n>=0} (x^n/n!) * A(x)^(2^n) dx.
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%I #9 Feb 25 2021 13:47:46

%S 1,1,2,7,38,295,3116,43609,801878,19525927,642957596,29433945337,

%T 1919020604078,180522951990631,24551166096301316,4811787151263424969,

%U 1353567127686626407766,544838534253508765596535,313088583104479511868150716,256427177192177984741354942569

%N E.g.f. A(x) satisfies: A(x) = 1 + Integral Sum_{n>=0} (x^n/n!) * A(x)^(2^n) dx.

%F E.g.f. A(x) satisfies:

%F (1) A(x) = 1 + Integral Sum_{n>=0} (x^n/n!) * A(x)^(2^n) dx.

%F (2) A(x) = 1 + Integral Sum_{n>=0} (log(A(x))^n/n!) * exp(2^n*x) dx.

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 38*x^4/4! + 295*x^5/5! + 3116*x^6/6! + 43609*x^7/7! + 801878*x^8/8! + 19525927*x^9/9! + 642957596*x^10/10! + ...

%e where

%e d/dx A(x) = A(x) + x*A(x)^2 + x^2*A(x)^(2^2)/2! + x^3*A(x)^(2^3)/3! + x^4*A(x)^(2^4)/4! + x^5*A(x)^(2^5)/5! + ...

%e also

%e d/dx A(x) = exp(x) + log(A(x))*exp(2*x) + log(A(x))^2*exp(2^2*x)/2! + log(A(x))^3*exp(2^3*x)/3! + log(A(x))^4*exp(2^4*x)/4! + ...

%e RELATED SERIES.

%e log(A(x)) = x + x^2/2! + 3*x^3/3! + 16*x^4/4! + 129*x^5/5! + 1456*x^6/6! + 22167*x^7/7! + 448666*x^8/8! + 12104541*x^9/9! + 441271816*x^10/10! + ...

%e The table of coefficients of x^k/k! in A(x)^(2^n) begins:

%e n=1: [1, 2, 6, 26, 156, 1250, 13032, 174242, ...];

%e n=2: [1, 4, 20, 124, 944, 8740, 97664, 1312300, ...];

%e n=3: [1, 8, 72, 728, 8256, 104840, 1488768, 23629304, ...];

%e n=4: [1, 16, 272, 4912, 94208, 1918480, 41474816, 951762544, ...];

%e n=5: [1, 32, 1056, 35936, 1261056, 45631520, 1702589952, ...];

%e n=6: [1, 64, 4160, 274624, 18412544, 1253769280, 86705859584, ...]; ...

%o (PARI) {a(n) = my(A=1+x +x*O(x^n)); for(i=1,n,

%o A = 1 + intformal( sum(m=0,n, x^m/m! * A^(2^m) +x*O(x^n)) ));

%o n!*polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n) = my(A=1+x +x*O(x^n)); for(i=1,n,

%o A = 1 + intformal( sum(m=0,n, log(A)^m/m! * exp(2^m*x +x*O(x^n))) ));

%o n!*polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 24 2021