A356773 E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} ( x^n + A(x) )^n * x^n / n!.

1, 1, 5, 22, 197, 2076, 29527, 477394, 9248745, 204340600, 5111234891, 142148945214, 4362830874877, 146338813894612, 5328688224075231, 209295914833477546, 8821420994034588113, 397128156446044087536, 19019218255697847951955, 965527468715744517674998

Offset: 0

Author

Paul D. Hanna, Aug 27 2022

Keywords

• nonn

Comments

More generally, the following sums are equal:

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

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

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

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 22*x^3/3! + 197*x^4/4! + 2076*x^5/5! + 29527*x^6/6! + 477394*x^7/7! + 9248745*x^8/8! + 204340600*x^9/9! + 5111234891*x^10/10! + ...
where
A(x) = 1 + (x + A(x))*x + (x^2 + A(x))^2*x^2/2! + (x^3 + A(x))^3*x^3/3! + (x^4 + A(x))^4*x^4/4! + (x^5 + A(x))^5*x^5/5! + ... + (x^n + A(x))^n*x^n/n! + ...
also
A(x) = exp(x*A(x)) + x^2*exp(x^2*A(x)) + x^6*exp(x^3*A(x))/2! + x^12*exp(x^4*A(x))/3! + x^20*exp(x^5*A(x))/4! + x^30*exp(x^6*A(x))/5! + ... +  x^(n*(n+1))*exp(x^(n+1)*A(x))/n! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 173*x^4/4! + 1956*x^5/5! + 27007*x^6/6! + 453874*x^7/7! + 8790105*x^8/8! + 195462136*x^9/9! + 4899670811*x^10/10! + ...
log(A(x)) = x + 4*x^2/2! + 9*x^3/3! + 88*x^4/4! + 905*x^5/5! + 12606*x^6/6! + 189217*x^7/7! + 3600472*x^8/8! + 78839217*x^9/9! + 1944056890*x^10/10! + ...
SPECIFIC VALUES.
A(x = 1/4) = 1.5376989442827462484156603674393740195...
A(x = 1/3) = 2.2880218830072453104841119982317247920...
A(x = 0.4) diverges.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A356772, A108459, A326090, A326091, A326261, A326009.

Programs

PARI

/* A(x) = Sum_{n>=0} ( x^n + A(x) )^n * x^n / n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, (x^m + A +x*O(x^n))^m*x^m/m! )); n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

PARI

/* A(x) = Sum_{n>=0} x^(n*(n+1)) * exp(x^(n+1)*A(x))/n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0,sqrtint(n), x^(m*(m+1)) * exp( x^(m+1)*A +x*O(x^n)) / m! )); n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

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

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

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

a(n) ~ c * d^n * n! / n^(3/2), where d = 2.88676786838244269... and c = 1.26061634709684... - Vaclav Kotesovec, Jul 03 2025