A202999 Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (A(x)^n + 1)^n * x^n/n!.

1, 2, 8, 80, 1392, 34352, 1108576, 44340704, 2119928320, 118111781888, 7524579815424, 540141484897280, 43182173208678400, 3808622859938226176, 367715812648914460672, 38610662734158029938688, 4384921058923036753723392, 536091721631513000647393280

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following.

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

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

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

Example

E.g.f.: A(x) = 1 + 2*x + 8*x^2/2! + 80*x^3/3! + 1392*x^4/4! + 34352*x^5/5! +...
where the e.g.f. satisfies following series identity:
A(x) = 1 + (A(x)+1)*x + (A(x)^2+1)^2*x^2/2! + (A(x)^3+1)^3*x^3/3! + (A(x)^4+1)^4*x^4/4! +...
A(x) = exp(x) + A(x)*exp(x*A(x))*x + A(x)^4*exp(x*A(x)^2)*x^2/2! + A(x)^9*exp(x*A(x)^3)*x^3/3! + A(x)^16*exp(x*A(x)^4)*x^4/4! +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (A^k+1+x*O(x^n))^k*x^k/k!)); n!*polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, A^(k^2)*exp(A^k*x+x*O(x^n))*x^k/k!)); n!*polcoeff(A, n)}

Crossrefs

Cf. A361053, A361054, A361055, A361057, A203013.

Cf. A361540.

Sequence in context: A230880 A361425 A214689 * A308088 A130530 A134529

Adjacent sequences: A202996 A202997 A202998 * A203000 A203001 A203002

Keyword

nonn

Author

Paul D. Hanna, Dec 27 2011

Status

approved