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

%I #10 Nov 15 2023 03:54:53

%S 1,1,1,3,12,68,473,3998,39327,443599,5629807,79486044,1235018598,

%T 20946691457,385025599130,7624623236381,161823815625933,

%U 3664505951884255,88189911547566082,2247691180645108608,60480432646998315279,1713328345952593367876,50970518521542636421145

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

%C (a(n)/(n-1)!)^(1/n) tends to 1.42011... - _Vaclav Kotesovec_, Nov 15 2023

%H Paul D. Hanna, <a href="/A366228/b366228.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.

%F (1) A(x) = 1 + Integral A(x)^A(x) dx.

%F (2) A(x) = exp( Integral A(x)^(A(x) - 1) dx ).

%F (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.

%e 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 where A(x) = 1 + Integral A(x)^A(x) dx.

%e RELATED SERIES.

%e 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 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 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 (PARI) {a(n) = my(A=1); for(i=0, n, A = 1 + intformal( A^A +x*O(x^n) ) ); n!*polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (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 for(n=0, 30, print1(a(n), ", "))

%Y Cf. A194786, A176118.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Nov 13 2023