A356772 E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} ( x^n + x*A(x) )^n / n!.

1, 2, 5, 34, 329, 3716, 55777, 1010206, 21187049, 511352272, 13929248861, 422450642054, 14129873671069, 516664310959720, 20503766568423881, 877759284120870526, 40321132468408643153, 1978363648482263649728, 103262474042895179595061, 5713315282015940379009862

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 = x*A(x), r = 1.

Examples

			E.g.f.: A(x) = 1 + 2*x + 5*x^2/2! + 34*x^3/3! + 329*x^4/4! + 3716*x^5/5! + 55777*x^6/6! + 1010206*x^7/7! + 21187049*x^8/8! + 511352272*x^9/9! + 13929248861*x^10/10! + ...
where
A(x) = 1 + (x + x*A(x)) + (x^2 + x*A(x))^2/2! + (x^3 + x*A(x))^3/3! + (x^4 + x*A(x))^4/4! + (x^5 + x*A(x))^5/5! + ... + (x^n + x*A(x))^n/n! + ...
also
A(x) = exp(x*A(x)) + x*exp(x^2*A(x)) + x^4*exp(x^3*A(x))/2! + x^9*exp(x^4*A(x))/3! + x^16*exp(x^5*A(x))/4! + x^25*exp(x^6*A(x))/5! + ... +  + x^(n^2)*exp(x^(n+1)*A(x))/n! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 5*x^2/2! + 28*x^3/3! + 269*x^4/4! + 3356*x^5/5! + 50257*x^6/6! + 915076*x^7/7! + 19427753*x^8/8! + 471310984*x^9/9! + 12892968701*x^10/10! + ...
log(A(x)) = 2*x + x^2/2! + 20*x^3/3! + 126*x^4/4! + 1314*x^5/5! + 20460*x^6/6! + 347906*x^7/7! + 7181944*x^8/8! + 170606106*x^9/9! + 4577504760*x^10/10! + ...
SPECIFIC VALUES.
A(x = 1/4) = 1.8854569251645435475372616427080...
A(x = 0.3) = 2.4910587821818158559566392662113...
A(x = 1/3) diverges.
		

Links

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

Crossrefs

Cf. A356773, A108459, A326090, A326091, A326261, A326009.

Programs

PARI

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

PARI

/* A(x) = Sum_{n>=0} x^(n^2) * exp(x^(n+1)*A(x))/n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0,sqrtint(n), x^(m^2) * 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 + x*A(x) )^n / n!.

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

a(n) ~ c * d^n * n! / n^(3/2), where d = 3.14614463757985697... and c = 1.454175198420213... - Vaclav Kotesovec, Jul 03 2025