A385057 E.g.f. satisfies A(x) = exp( Sum_{n>=1} (Integral A(x)^n dx)^n / n ).

1, 1, 3, 18, 173, 2368, 43025, 991070, 28030227, 950818494, 37995695979, 1763496545502, 93967776822477, 5692538342703978, 388772833646583213, 29711642817587338986, 2524166742181661207511, 236956380718244960455206, 24446253183753019240769463, 2757979540962272093582650734, 338712272097534292284500861745

Offset: 0

Author

Paul D. Hanna, Aug 19 2025

Keywords

• nonn
• new

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 18*x^3/3! + 173*x^4/4! + 2368*x^5/5! + 43025*x^6/6! + 991070*x^7/7! + 28030227*x^8/8! + 950818494*x^9/9! + 37995695979*x^10/10! + ...
where
A(x) = exp( (Integral A(x) dx) + (Integral A(x)^2 dx)^2/2 + (Integral A(x)^3 dx)^3/3 + (Integral A(x)^4 dx)^4/4 + ... ).
Also,
A'(x) = A(x)^2 + A(x)^3*(Integral A(x)^2 dx) + A(x)^4*(Integral A(x)^3 dx)^2 + A(x)^5*(Integral A(x)^4 dx)^3 + ...
RELATED SERIES.
log(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 104*x^4/4! + 1437*x^5/5! + 26642*x^6/6! + 629127*x^7/7! + ... + A268294(n)*x^n/n! + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..201

Crossrefs

Cf. A268294 (log).

Programs

PARI

{a(n) = my(A = 1 + x +x*O(x^n)); for(i=0, n+1, A = exp( sum(m=1, n+1, intformal(A^m)^m/m ) ) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

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

(1) A(x) = exp( Sum_{n>=1} (Integral A(x)^n dx)^n / n ).

(2) A'(x) = Sum_{n>=1} A(x)^(n+1) * (Integral A(x)^n dx)^(n-1).

(3) A(x) = exp(B(x)), where B(x) is the e.g.f. of A268294.