A385907 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} (1/(1-x)^n - A(x))^n / n!.

1, 1, 3, 23, 352, 8514, 283631, 12112806, 635825408, 39879146040, 2927184461577, 247446468028660, 23783152246963000, 2571886518254945436, 310182327935629919849, 41412336266329078729672, 6081427517910705699609120, 976832191726100607511719504, 170781177018035313244639721409

Offset: 0

Author

Paul D. Hanna, Jul 24 2025

Keywords

• nonn

Comments

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 = 1/(1-x) with p = -A(x), r = 1.

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 23*x^3/3! + 352*x^4/4! + 8514*x^5/5! + 283631*x^6/6! + 12112806*x^7/7! + 635825408*x^8/8! + 39879146040*x^9/9! + 2927184461577*x^10/10! + ...
where
1 = 1 + (1/(1-x) - A(x)) + (1/(1-x)^2 - A(x))^2/2! + (1/(1-x)^3 - A(x))^3/3! + (1/(1-x)^4 - A(x))^4/4! + (1/(1-x)^5 - A(x))^5/5! + (1/(1-x)^6 - A(x))^6/6! + ...
also
1 = exp(-A(x)) + 1/(1-x)*exp(-A(x)/(1-x)) + 1/(1-x)^4*exp(-A(x)/(1-x)^2)/2! + 1/(1-x)^9*exp(-A(x)/(1-x)^3)/3! + 1/(1-x)^16*exp(-A(x)/(1-x)^4)/4! + 1/(1-x)^25*exp(-A(x)/(1-x)^5)/5! + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..260

Crossrefs

Cf. A326097.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (1/(1-x)^m - Ser(A))^m/m! ), #A-1) ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} (1/(1-x)^n - A(x))^n / n!.

(2) 1 = Sum_{n>=0} 1/(1-x)^(n^2) * exp(-A(x)/(1-x)^n) / n!.