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Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 3)^n * x^n/n!.
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%I #10 Mar 20 2023 15:00:00

%S 1,4,24,328,8480,316064,15448000,940586560,68773511680,5883198833152,

%T 577566163260416,64112172571384832,7953180924959641600,

%U 1092205827724943429632,164769061745517773774848,27131359440809990936141824,4850231804845681441360707584,937096082325039305880612503552

%N Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 3)^n * x^n/n!.

%H Paul D. Hanna, <a href="/A361054/b361054.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined as follows.

%F (1) A(x) = Sum_{n>=0} (A(x)^n + 3)^n * x^n/n!.

%F (2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(3*x*A(x)^n) * x^n/n!.

%F a(n) = 0 (mod 4) for n > 0.

%F a(n) = Sum_{k=0..n} A361540(n,k) * 3^k. - _Paul D. Hanna_, Mar 20 2023

%e E.g.f.: A(x) = 1 + 4*x + 24*x^2/2! + 328*x^3/3! + 8480*x^4/4! + 316064*x^5/5! + 15448000*x^6/6! + 940586560*x^7/7! + 68773511680*x^8/8! +...

%e where the e.g.f. satisfies the following series identity:

%e A(x) = 1 + (A(x) + 3)*x + (A(x)^2 + 3)^2*x^2/2! + (A(x)^3 + 3)^3*x^3/3! + (A(x)^4 + 3)^4*x^4/4! + ... + (A(x)^n + 3)^n * x^n/n! + ...

%e and

%e A(x) = exp(3*x) + A(x)*exp(3*x*A(x))*x + A(x)^4*exp(3*x*A(x)^2)*x^2/2! + A(x)^9*exp(3*x*A(x)^3)*x^3/3! + A(x)^16*exp(3*x*A(x)^4)*x^4/4! + ... + A(x)^(n^2) * exp(3*x*A(x)^n) * x^n/n! + ...

%o (PARI) /* E.g.f.: Sum_{n>=0} (A(x)^n + 3)^n * x^n/n! */

%o {a(n) = my(A = 1); for(i=1,n, A = sum(m=0, n, (A^m + 3 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}

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

%o (PARI) /* E.g.f.: Sum_{n>=0} A(x)^(n^2) * exp(3*x*A(x)^n) * x^n/n! */

%o {a(n) = my(A=1); for(i=1,n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(3*x*A^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}

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

%Y Cf. A202999, A361053, A361055, A361056, A361057, A203013.

%Y Cf. A361540.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 28 2023