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A361057 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (3*A(x)^n + 1)^n * x^n/n!.

%I A361057 #11 Mar 20 2023 15:06:05
%S A361057 1,4,40,1000,42208,2511904,194701888,18644964160,2128895802880,
%T A361057 282664859507200,42830926407126016,7299282818219035648,
%U A361057 1382930912338770866176,288548709643121903915008,65787364162207649519116288,16282501210870115738111156224,4350458941547832791800523653120
%N A361057 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (3*A(x)^n + 1)^n * x^n/n!.
%H A361057 Paul D. Hanna, <a href="/A361057/b361057.txt">Table of n, a(n) for n = 0..200</a>
%F A361057 E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined as follows.
%F A361057 (1) A(x) = Sum_{n>=0} (3*A(x)^n + 1)^n * x^n/n!.
%F A361057 (2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(x*A(x)^n) * 3^n * x^n/n!.
%F A361057 a(n) = 0 (mod 4) for n > 0.
%F A361057 a(n) = Sum_{k=0..n} A361540(n,k) * 3^(n-k). - _Paul D. Hanna_, Mar 20 2023
%e A361057 E.g.f.: A(x) = 1 + 4*x + 40*x^2/2! + 1000*x^3/3! + 42208*x^4/4! + 2511904*x^5/5! + 194701888*x^6/6! + 18644964160*x^7/7! + 2128895802880*x^8/8! +...
%e A361057 where the e.g.f. satisfies the following series identity:
%e A361057 A(x) = 1 + (3*A(x) + 1)*x + (3*A(x)^2 + 1)^2*x^2/2! + (3*A(x)^3 + 1)^3*x^3/3! + (3*A(x)^4 + 1)^4*x^4/4! + ... + (3*A(x)^n + 1)^n * x^n/n! + ...
%e A361057 and
%e A361057 A(x) = exp(x) + A(x)*exp(x*A(x))*3*x + A(x)^4*exp(x*A(x)^2)*3^2*x^2/2! + A(x)^9*exp(x*A(x)^3)*3^3*x^3/3! + A(x)^16*exp(x*A(x)^4)*3^4*x^4/4! + ... + A(x)^(n^2) * exp(x*A(x)^n) * 3^n * x^n/n! + ...
%o A361057 (PARI) /* E.g.f.: Sum_{n>=0} (3*A(x)^n + 1)^n * x^n/n! */
%o A361057 {a(n) = my(A = 1); for(i=1,n, A = sum(m=0, n, (3*A^m + 1 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}
%o A361057 for(n=0, 20, print1(a(n), ", "))
%o A361057 (PARI) /* E.g.f.: Sum_{n>=0} A(x)^(n^2) * exp(x*A(x)^n) * 3^n * x^n/n! */
%o A361057 {a(n) = my(A=1); for(i=1,n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(x*A^m +x*O(x^n)) * 3^m * x^m/m! )); n!*polcoeff(A, n)}
%o A361057 for(n=0, 20, print1(a(n), ", "))
%Y A361057 Cf. A202999, A361053, A361054, A361055, A361056, A203013.
%Y A361057 Cf. A361540.
%K A361057 nonn
%O A361057 0,2
%A A361057 _Paul D. Hanna_, Feb 28 2023