A386645 E.g.f. A(x) satisfies A(x) = Sum_{n>=0} ( A(x)^n + log(A(x)) )^n * x^n / n!.

1, 1, 5, 46, 665, 13416, 350227, 11254300, 430093617, 19067201056, 962456078051, 54518610032844, 3425698051345561, 236531783119320352, 17805560350371525747, 1451679746048430507676, 127461102015439274388833, 11993733613161390301999680, 1204348142606115777871016899, 128572080570190461521783550988

Offset: 0

Author

Paul D. Hanna, Aug 08 2025

Keywords

• nonn

Comments

It appears that for n > 6, a(n) (mod 6) equals [1, 4, 3, 4, 5, 0] repeating.

In general, the following sums are equal:

(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,

(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = A(x) with p = log(A(x)), r = x.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 46*x^3/3! + 665*x^4/4! + 13416*x^5/5! + 350227*x^6/6! + 11254300*x^7/7! + 430093617*x^8/8! + 19067201056*x^9/9! + ...
where
A(x) = A(x)^x + A(x)*A(x)^(x*A(x))*x + A(x)^4*A(x)^(x*A(x)^2)*x^2/2! + A(x)^9*A(x)^(x*A(x)^3)*x^3/3! + A(x)^16*A(x)^(x*A(x)^4)*x^4/4! + ...
Also,
A(x) = 1 + (A(x) + log(A(x)))*x + (A(x)^2 + log(A(x)))^2*x^2/2! + (A(x)^3 + log(A(x)))^3*x^3/3! + (A(x)^4 + log(A(x)))^4*x^4/4! + ...
RELATED SERIES.
log(A(x)) = x + 4*x^2/2! + 33*x^3/3! + 460*x^4/4! + 9185*x^5/5! + 239406*x^6/6! + 7704403*x^7/7! + 295172872*x^8/8! + 13123492929*x^9/9! + 664403960890*x^10/10! + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..230

Crossrefs

Cf. A386644.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (Ser(A)^m + log(Ser(A)))^m * x^m/m! ), #A-1) ); n!*A[n+1]}
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) = Sum_{n>=0} A(x)^(n^2) * A(x)^(x*A(x)^n) * x^n / n!.

(2) A(x) = Sum_{n>=0} ( A(x)^n + log(A(x)) )^n * x^n / n!.