A361057 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (3*A(x)^n + 1)^n * x^n/n!.

1, 4, 40, 1000, 42208, 2511904, 194701888, 18644964160, 2128895802880, 282664859507200, 42830926407126016, 7299282818219035648, 1382930912338770866176, 288548709643121903915008, 65787364162207649519116288, 16282501210870115738111156224, 4350458941547832791800523653120

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} (3*A(x)^n + 1)^n * x^n/n!.

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

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

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

Example

E.g.f.: A(x) = 1 + 4*x + 40*x^2/2! + 1000*x^3/3! + 42208*x^4/4! + 2511904*x^5/5! + 194701888*x^6/6! + 18644964160*x^7/7! + 2128895802880*x^8/8! +...
where the e.g.f. satisfies the following series identity:
A(x) = 1 + (3*A(x) + 1)*x + (3*A(x)^2 + 1)^2*x^2/2! + (3*A(x)^3 + 1)^3*x^3/3! + (3*A(x)^4 + 1)^4*x^4/4! + ... + (3*A(x)^n + 1)^n * x^n/n! + ...
and
A(x) = exp(x) + A(x)*exp(x*A(x))*3*x + A(x)^4*exp(x*A(x)^2)*3^2*x^2/2! + A(x)^9*exp(x*A(x)^3)*3^3*x^3/3! + A(x)^16*exp(x*A(x)^4)*3^4*x^4/4! + ... + A(x)^(n^2) * exp(x*A(x)^n) * 3^n * x^n/n! + ...

Prog

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

Crossrefs

Cf. A202999, A361053, A361054, A361055, A361056, A203013.

Cf. A361540.

Sequence in context: A012957 A012977 A321526 * A013108 A173945 A111846

Adjacent sequences: A361054 A361055 A361056 * A361058 A361059 A361060

Keyword

nonn

Author

Paul D. Hanna, Feb 28 2023

Status

approved