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A361053 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 2)^n * x^n/n!.

%I A361053 #9 Mar 20 2023 14:58:42
%S A361053 1,3,15,180,3933,122778,5024727,255694050,15594132825,1110807585090,
%T A361053 90665847445059,8355178654847874,859198582766876661,
%U A361053 97668423691415577666,12177783763614287432847,1654751006054203510476882,243720706148230009547388465,38730619011753683906970442626
%N A361053 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 2)^n * x^n/n!.
%H A361053 Paul D. Hanna, <a href="/A361053/b361053.txt">Table of n, a(n) for n = 0..200</a>
%F A361053 E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined as follows.
%F A361053 (1) A(x) = Sum_{n>=0} (A(x)^n + 2)^n * x^n/n!.
%F A361053 (2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(2*x*A(x)^n) * x^n/n!.
%F A361053 a(n) = 0 (mod 3) for n > 0.
%F A361053 a(n) = Sum_{k=0..n} A361540(n,k) * 2^k. - _Paul D. Hanna_, Mar 20 2023
%e A361053 E.g.f.: A(x) = 1 + 3*x + 15*x^2/2! + 180*x^3/3! + 3933*x^4/4! + 122778*x^5/5! + 5024727*x^6/6! + 255694050*x^7/7! + 15594132825*x^8/8! +...
%e A361053 where the e.g.f. satisfies the following series identity:
%e A361053 A(x) = 1 + (A(x) + 2)*x + (A(x)^2 + 2)^2*x^2/2! + (A(x)^3 + 2)^3*x^3/3! + (A(x)^4 + 2)^4*x^4/4! + ... + (A(x)^n + 2)^n * x^n/n! + ...
%e A361053 and
%e A361053 A(x) = exp(2*x) + A(x)*exp(2*x*A(x))*x + A(x)^4*exp(2*x*A(x)^2)*x^2/2! + A(x)^9*exp(2*x*A(x)^3)*x^3/3! + A(x)^16*exp(2*x*A(x)^4)*x^4/4! + ... + A(x)^(n^2) * exp(2*x*A(x)^n) * x^n/n! + ...
%o A361053 (PARI) /* E.g.f.: Sum_{n>=0} (A(x)^n + 2)^n * x^n/n! */
%o A361053 {a(n) = my(A = 1); for(i=1,n, A = sum(m=0, n, (A^m + 2 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}
%o A361053 for(n=0, 20, print1(a(n), ", "))
%o A361053 (PARI) /* E.g.f.: Sum_{n>=0} A(x)^(n^2) * exp(2*x*A(x)^n) * x^n/n! */
%o A361053 {a(n) = my(A=1); for(i=1,n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(2*x*A^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}
%o A361053 for(n=0, 20, print1(a(n), ", "))
%Y A361053 Cf. A202999, A361054, A361055, A361056, A361057, A203013.
%Y A361053 Cf. A361540.
%K A361053 nonn
%O A361053 0,2
%A A361053 _Paul D. Hanna_, Feb 28 2023