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).
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = exp(x) * A(x^2*exp(x)).
(2) A(x) = exp( 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) = exp(L(x)) where L(x) = x + L(x^2*exp(x)) is the e.g.f of A369091.
(4) A(x) = G(x)/x where G(x) = G(x^2*exp(x))/x is the e.g.f. of A369090.
a(n) = A369090(n+1)/(n+1) for n >= 0.
EXAMPLE
E.g.f.: 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! + 15924025*x^9/9! + ...
RELATED SERIES.
The expansion of A(x^2*exp(x)) begins
exp(-x) * A(x) = A(x^2*exp(x)) = 1 + 2*x^2/2! + 6*x^3/3! + 48*x^4/4! + 380*x^5/5! + 3750*x^6/6! + + 42882*x^7/7! + 576296*x^8/8! + ...
The logarithm of e.g.f. A(x) equals L(x) where L(x) = x + L(x^2*exp(x)),
L(x) = x + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2190*x^6/6! + 21882*x^7/7! + 268856*x^8/8! + ... + A369091(n)*x^n/n! + ...
PROG
(PARI) {a(n) = my(A=1+x, X = x + x*O(x^n)); for(i=1, n, A = exp(X) * subst(A, x, x^2*exp(X)) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 29 2024
STATUS
approved