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

1, 1, 1, 3, 12, 68, 473, 3998, 39327, 443599, 5629807, 79486044, 1235018598, 20946691457, 385025599130, 7624623236381, 161823815625933, 3664505951884255, 88189911547566082, 2247691180645108608, 60480432646998315279, 1713328345952593367876, 50970518521542636421145

Offset: 0

Author

Paul D. Hanna, Nov 13 2023

Keywords

• nonn

Comments

(a(n)/(n-1)!)^(1/n) tends to 1.42011... - Vaclav Kotesovec, Nov 15 2023

Examples

			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! + ...
where A(x) = 1 + Integral A(x)^A(x) dx.
RELATED SERIES.
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! + ...
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! + ...
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! + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A194786, A176118.

Programs

PARI

{a(n) = my(A=1); for(i=0, n, A = 1 + intformal( A^A +x*O(x^n) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n) = my(A=1); for(i=0, n, A = exp( intformal( A^(A-1) +x*O(x^n) ) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

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

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

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

(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.

(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