A369551 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*exp(x) * A(x^2*exp(x)).

1, 1, 2, 9, 52, 365, 3126, 33607, 434120, 6397785, 104813290, 1881831611, 36703128012, 773468319637, 17544261523166, 427299522260535, 11158470652237456, 311944165793916977, 9313905287778153426, 296051128664550195763, 9979462242106491507860, 355292353569342771519021

Offset

0,3

Comments

Limit (a(n)/n!)^(1/n) = 1/w where w*exp(w) = 1 and w = LambertW(1) = 0.567143290409783872999968... (cf. A030178).

Links

Table of n, a(n) for n=0..21.

Formula

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

(1) A(x) = 1 + x*exp(x) * A(x^2*exp(x)).

(2) A(x) = (1/x) * Sum_{n>=0} F(n), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.

(3) A(x) = log(G(x)) / x where G(x) = exp(x) * G(x^2*exp(x)) is the e.g.f. of A369550.

(4) A(x) = L(x)/x where L(x) = x + L(x^2*exp(x)) is the e.g.f of A369091.

a(n) = A369091(n+1)/(n+1) for n >= 0.

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 365*x^5/5! + 3126*x^6/6! + 33607*x^7/7! + 434120*x^8/8! + 6397785*x^9/9! + 104813290*x^10/10! + ...
RELATED SERIES.
The expansion of A(x^2*exp(x)) begins
A(x^2*exp(x)) = 1 + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2550*x^6/6! + 29442*x^7/7! + 386456*x^8/8! + ...
where A(x) = 1 + x*exp(x) * A(x^2*exp(x)).
The expansion of exp(x*A(x)) is the e.g.f. of A369550, which begins
exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! + 6901*x^6/6! + 79045*x^7/7! + 1049385*x^8/8! + ... + A369550(n)*x^n/n! + ...

Prog

(PARI) {a(n) = my(A=1+x, X = x + x*O(x^n)); for(i=1, n, A = 1 + x*exp(X) * subst(A, x, x^2*exp(X)) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A369090, A369091, A369550.

Sequence in context: A355789 A231494 A006152 * A143508 A052882 A248440

Adjacent sequences: A369548 A369549 A369550 * A369552 A369553 A369554

Keyword

nonn

Author

Paul D. Hanna, Jan 29 2024

Status

approved