A361053 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 2)^n * x^n/n!.

1, 3, 15, 180, 3933, 122778, 5024727, 255694050, 15594132825, 1110807585090, 90665847445059, 8355178654847874, 859198582766876661, 97668423691415577666, 12177783763614287432847, 1654751006054203510476882, 243720706148230009547388465, 38730619011753683906970442626

Offset: 0

Author

Paul D. Hanna, Feb 28 2023

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + 3*x + 15*x^2/2! + 180*x^3/3! + 3933*x^4/4! + 122778*x^5/5! + 5024727*x^6/6! + 255694050*x^7/7! + 15594132825*x^8/8! +...
where the e.g.f. satisfies the following series identity:
A(x) = 1 + (A(x) + 2)*x + (A(x)^2 + 2)^2*x^2/2! + (A(x)^3 + 2)^3*x^3/3! + (A(x)^4 + 2)^4*x^4/4! + ... + (A(x)^n + 2)^n * x^n/n! + ...
and
A(x) = exp(2*x) + A(x)*exp(2*x*A(x))*x + A(x)^4*exp(2*x*A(x)^2)*x^2/2! + A(x)^9*exp(2*x*A(x)^3)*x^3/3! + A(x)^16*exp(2*x*A(x)^4)*x^4/4! + ... + A(x)^(n^2) * exp(2*x*A(x)^n) * x^n/n! + ...
		

Links

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

Crossrefs

Cf. A202999, A361054, A361055, A361056, A361057, A203013.

Cf. A361540.

Programs

PARI

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

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

a(n) = 0 (mod 3) for n > 0.

a(n) = Sum_{k=0..n} A361540(n,k) * 2^k. - Paul D. Hanna, Mar 20 2023