A361055 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 4)^n * x^n/n!.

1, 5, 35, 530, 15645, 673100, 37951975, 2668045700, 225591547225, 22347122264900, 2543582111665875, 327736278022956500, 47245927138947731125, 7548695252947520166500, 1326483608786914301185375, 254733442821907695977652500, 53175506363950820566794680625, 12012490474019349963485905242500

Offset: 0

Author

Paul D. Hanna, Feb 28 2023

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + 5*x + 35*x^2/2! + 530*x^3/3! + 15645*x^4/4! + 673100*x^5/5! + 37951975*x^6/6! + 2668045700*x^7/7! + 225591547225*x^8/8! +...
where the e.g.f. satisfies the following series identity:
A(x) = 1 + (A(x) + 4)*x + (A(x)^2 + 4)^2*x^2/2! + (A(x)^3 + 4)^3*x^3/3! + (A(x)^4 + 4)^4*x^4/4! + ... + (A(x)^n + 4)^n * x^n/n! + ...
and
A(x) = exp(4*x) + A(x)*exp(4*x*A(x))*x + A(x)^4*exp(4*x*A(x)^2)*x^2/2! + A(x)^9*exp(4*x*A(x)^3)*x^3/3! + A(x)^16*exp(4*x*A(x)^4)*x^4/4! + ... + A(x)^(n^2) * exp(4*x*A(x)^n) * x^n/n! + ...
		

Links

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

Crossrefs

Cf. A202999, A361053, A361054, A361056, A361057, A203013.

Cf. A361540.

Programs

PARI

/* E.g.f.: Sum_{n>=0} (A(x)^n + 4)^n * x^n/n! */
{a(n) = my(A = 1); for(i=1,n, A = sum(m=0, n, (A^m + 4 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

PARI

/* E.g.f.: Sum_{n>=0} A(x)^(n^2) * exp(4*x*A(x)^n) * x^n/n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0, n, (A +x*O(x^n))^(m^2) * exp(4*x*A^m +x*O(x^n)) * x^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! may be defined as follows.

(1) A(x) = Sum_{n>=0} (A(x)^n + 4)^n * x^n/n!.

(2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(4*x*A(x)^n) * x^n/n!.

a(n) = 0 (mod 5) for n > 0.

a(n) = Sum_{k=0..n} A361540(n,k) * 4^k. - Paul D. Hanna, Mar 20 2023