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A361055 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 4)^n * x^n/n!.

%I A361055 #9 Mar 20 2023 15:00:46
%S A361055 1,5,35,530,15645,673100,37951975,2668045700,225591547225,
%T A361055 22347122264900,2543582111665875,327736278022956500,
%U A361055 47245927138947731125,7548695252947520166500,1326483608786914301185375,254733442821907695977652500,53175506363950820566794680625,12012490474019349963485905242500
%N A361055 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 4)^n * x^n/n!.
%H A361055 Paul D. Hanna, <a href="/A361055/b361055.txt">Table of n, a(n) for n = 0..200</a>
%F A361055 E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined as follows.
%F A361055 (1) A(x) = Sum_{n>=0} (A(x)^n + 4)^n * x^n/n!.
%F A361055 (2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(4*x*A(x)^n) * x^n/n!.
%F A361055 a(n) = 0 (mod 5) for n > 0.
%F A361055 a(n) = Sum_{k=0..n} A361540(n,k) * 4^k. - _Paul D. Hanna_, Mar 20 2023
%e A361055 E.g.f.: A(x) = 1 + 5*x + 35*x^2/2! + 530*x^3/3! + 15645*x^4/4! + 673100*x^5/5! + 37951975*x^6/6! + 2668045700*x^7/7! + 225591547225*x^8/8! +...
%e A361055 where the e.g.f. satisfies the following series identity:
%e A361055 A(x) = 1 + (A(x) + 4)*x + (A(x)^2 + 4)^2*x^2/2! + (A(x)^3 + 4)^3*x^3/3! + (A(x)^4 + 4)^4*x^4/4! + ... + (A(x)^n + 4)^n * x^n/n! + ...
%e A361055 and
%e A361055 A(x) = exp(4*x) + A(x)*exp(4*x*A(x))*x + A(x)^4*exp(4*x*A(x)^2)*x^2/2! + A(x)^9*exp(4*x*A(x)^3)*x^3/3! + A(x)^16*exp(4*x*A(x)^4)*x^4/4! + ... + A(x)^(n^2) * exp(4*x*A(x)^n) * x^n/n! + ...
%o A361055 (PARI) /* E.g.f.: Sum_{n>=0} (A(x)^n + 4)^n * x^n/n! */
%o A361055 {a(n) = my(A = 1); for(i=1,n, A = sum(m=0, n, (A^m + 4 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}
%o A361055 for(n=0, 20, print1(a(n), ", "))
%o A361055 (PARI) /* E.g.f.: Sum_{n>=0} A(x)^(n^2) * exp(4*x*A(x)^n) * x^n/n! */
%o A361055 {a(n) = my(A=1); for(i=1,n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(4*x*A^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}
%o A361055 for(n=0, 20, print1(a(n), ", "))
%Y A361055 Cf. A202999, A361053, A361054, A361056, A361057, A203013.
%Y A361055 Cf. A361540.
%K A361055 nonn
%O A361055 0,2
%A A361055 _Paul D. Hanna_, Feb 28 2023