A361054 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 3)^n * x^n/n!.

1, 4, 24, 328, 8480, 316064, 15448000, 940586560, 68773511680, 5883198833152, 577566163260416, 64112172571384832, 7953180924959641600, 1092205827724943429632, 164769061745517773774848, 27131359440809990936141824, 4850231804845681441360707584, 937096082325039305880612503552

Offset

0,2

Links

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

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 + 3)^n * x^n/n!.

(2) A(x) = Sum_{n>=0} A(x)^(n^2) * exp(3*x*A(x)^n) * x^n/n!.

a(n) = 0 (mod 4) for n > 0.

a(n) = Sum_{k=0..n} A361540(n,k) * 3^k. - Paul D. Hanna, Mar 20 2023

Example

E.g.f.: A(x) = 1 + 4*x + 24*x^2/2! + 328*x^3/3! + 8480*x^4/4! + 316064*x^5/5! + 15448000*x^6/6! + 940586560*x^7/7! + 68773511680*x^8/8! +...
where the e.g.f. satisfies the following series identity:
A(x) = 1 + (A(x) + 3)*x + (A(x)^2 + 3)^2*x^2/2! + (A(x)^3 + 3)^3*x^3/3! + (A(x)^4 + 3)^4*x^4/4! + ... + (A(x)^n + 3)^n * x^n/n! + ...
and
A(x) = exp(3*x) + A(x)*exp(3*x*A(x))*x + A(x)^4*exp(3*x*A(x)^2)*x^2/2! + A(x)^9*exp(3*x*A(x)^3)*x^3/3! + A(x)^16*exp(3*x*A(x)^4)*x^4/4! + ... + A(x)^(n^2) * exp(3*x*A(x)^n) * x^n/n! + ...

Prog

(PARI) /* E.g.f.: Sum_{n>=0} (A(x)^n + 3)^n * x^n/n! */
{a(n) = my(A = 1); for(i=1, n, A = sum(m=0, n, (A^m + 3 +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(3*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(3*x*A^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A202999, A361053, A361055, A361056, A361057, A203013.

Cf. A361540.

Sequence in context: A219118 A005756 A206239 * A196687 A185457 A024250

Adjacent sequences: A361051 A361052 A361053 * A361055 A361056 A361057

Keyword

nonn

Author

Paul D. Hanna, Feb 28 2023

Status

approved