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A386644 E.g.f. A(x) satisfies A(x) = Sum_{n>=0} (A(x)^n + x)^n * x^n / n!.

%I A386644 #11 Aug 09 2025 14:15:31
%S A386644 1,1,5,34,437,7996,191497,5679178,200959929,8269303384,388201586381,
%T A386644 20486491855534,1201171090068325,77504136748838164,
%U A386644 5460029344935045441,417185040885539939506,34377042102420367770353,3040184386700809821194416,287334696971272926921192469,28915390444625255004763736278
%N A386644 E.g.f. A(x) satisfies A(x) = Sum_{n>=0} (A(x)^n + x)^n * x^n / n!.
%C A386644 It appears that for n > 6, a(n) (mod 6) equals [1, 4, 3, 4, 5, 4] repeating.
%C A386644 In general, the following sums are equal:
%C A386644 (C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
%C A386644 (C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C A386644 here, q = A(x) with p = x, r = x.
%H A386644 Paul D. Hanna, <a href="/A386644/b386644.txt">Table of n, a(n) for n = 0..200</a>
%F A386644 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A386644 (1) A(x) = Sum_{n>=0} (A(x)^n + x)^n * x^n / n!.
%F A386644 (2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(x^2*A(x)^n) * x^n / n!.
%e A386644 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 34*x^3/3! + 437*x^4/4! + 7996*x^5/5! + 191497*x^6/6! + 5679178*x^7/7! + 200959929*x^8/8! + 8269303384*x^9/9! + ...
%e A386644 where
%e A386644 A(x) = 1 + (A(x) + x)*x + (A(x)^2 + x)^2*x^2/2! + (A(x)^3 + x)^3*x^3/3! + (A(x)^4 + x)^4*x^2/4! + (A(x)^5 + x)^5*x^5/5! + ...
%e A386644 Also,
%e A386644 A(x) = exp(x^2) + A(x)*exp(x^2*A(x))*x + A(x)^4*exp(x^2*A(x)^2)*x^2/2! + A(x)^9*exp(x^2*A(x)^3)*x^3/3! + A(x)^16*exp(x^2*A(x)^4)*x^4/4! + ...
%o A386644 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (Ser(A)^m + x)^m * x^m/m! ), #A-1) ); H=A; n!*A[n+1]}
%o A386644 for(n=0, 20, print1(a(n), ", "))
%Y A386644 Cf. A386645, A136516.
%K A386644 nonn
%O A386644 0,3
%A A386644 _Paul D. Hanna_, Aug 08 2025