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A366228 Expansion of e.g.f. A(x) satisfying A(x) = 1 + Integral A(x)^A(x) dx.

%I A366228 #14 Jul 26 2025 10:29:04
%S A366228 1,1,1,3,12,68,473,3998,39327,443599,5629807,79486044,1235018598,
%T A366228 20946691457,385025599130,7624623236381,161823815625933,
%U A366228 3664505951884255,88189911547566082,2247691180645108608,60480432646998315279,1713328345952593367876,50970518521542636421145
%N A366228 Expansion of e.g.f. A(x) satisfying A(x) = 1 + Integral A(x)^A(x) dx.
%C A366228 (a(n)/(n-1)!)^(1/n) tends to 1.42011... - _Vaclav Kotesovec_, Nov 15 2023
%H A366228 Paul D. Hanna, <a href="/A366228/b366228.txt">Table of n, a(n) for n = 0..300</a>
%F A366228 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A366228 (1) A(x) = 1 + Integral A(x)^A(x) dx.
%F A366228 (2) A(x) = exp( Integral A(x)^(A(x) - 1) dx ).
%F A366228 (3) A(x) = 1 + Series_Reversion( Integral 1/(1+x)^(1+x) dx ), where 1/(1+x)^(1+x) is the e.g.f. of A176118.
%F A366228 (4) A(x)^A(x) = 1/Sum_{n>=0} (1 - A(x))^n/n! * Product_{k=1..n} (k + A(x)-1) = A'(x). - _Paul D. Hanna_, Jul 25 2025
%e A366228 E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 12*x^4/4! + 68*x^5/5! + 473*x^6/6! + 3998*x^7/7! + 39327*x^8/8! + 443599*x^9/9! + 5629807*x^10/10! + ...
%e A366228 where A(x) = 1 + Integral A(x)^A(x) dx.
%e A366228 RELATED SERIES.
%e A366228 A(x)^A(x) = 1 + x + 3*x^2/2! + 12*x^3/3! + 68*x^4/4! + 473*x^5/5! + 3998*x^6/6! + 39327*x^7/7! + 443599*x^8/8! + ...
%e A366228 log(A(x)) = x + 2*x^3/3! + 3*x^4/4! + 32*x^5/5! + 155*x^6/6! + 1575*x^7/7! + 13573*x^8/8! + 160756*x^9/9! + 1938288*x^10/10! + ...
%e A366228 A(x)^(A(x) - 1) = 1 + 2*x^2/2! + 3*x^3/3! + 32*x^4/4! + 155*x^5/5! + 1575*x^6/6! + 13573*x^7/7! + ...
%o A366228 (PARI) {a(n) = my(A=1); for(i=0, n, A = 1 + intformal( A^A +x*O(x^n) ) ); n!*polcoeff(A, n)}
%o A366228 for(n=0, 30, print1(a(n), ", "))
%o A366228 (PARI) {a(n) = my(A=1); for(i=0, n, A = exp( intformal( A^(A-1) +x*O(x^n) ) ) ); n!*polcoeff(A, n)}
%o A366228 for(n=0, 30, print1(a(n), ", "))
%Y A366228 Cf. A194786, A176118.
%K A366228 nonn
%O A366228 0,4
%A A366228 _Paul D. Hanna_, Nov 13 2023