OFFSET
0,3
COMMENTS
Compare the e.g.f. to the identity:
exp(-x) = Sum_{n>=0} exp(n*x) * Product_{k=1..n} (1 - exp(k*x)).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..180
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.
FORMULA
E.g.f. A(x) satisfies: A(x) = exp(-x)*(2*G(x) - 1),
where G(x) = Sum_{n>=0} Product_{k=1..n} (exp(k*x) - 1) = e.g.f. of A158690.
a(n) ~ sqrt(2) * 12^(n+1) * (n!)^2 / Pi^(2*n+2). - Vaclav Kotesovec, May 05 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 85*x^3/3! + 1759*x^4/4! + 55621*x^5/5! +...
such that, by definition,
A(x) = 1 + exp(x) * (exp(x)-1) + exp(2*x) * (exp(x)-1)*(exp(2*x)-1)
+ exp(3*x) * (exp(x)-1)*(exp(2*x)-1)*(exp(3*x)-1)
+ exp(4*x) * (exp(x)-1)*(exp(2*x)-1)*(exp(3*x)-1)*(exp(4*x)-1) +...
The related e.g.f. of A158690 equals the series:
G(x) = 1 + (exp(x)-1) + (exp(x)-1)*(exp(2*x)-1)
+ (exp(x)-1)*(exp(2*x)-1)*(exp(3*x)-1)
+ (exp(x)-1)*(exp(2*x)-1)*(exp(3*x)-1)*(exp(4*x)-1) +...
or, more explicitly,
G(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1073*x^4/4! + 32671*x^5/5! +...
such that G(x) satisfies:
G(x) = (1 + exp(x)*A(x))/2.
PROG
(PARI) {a(n)=n!*polcoeff(sum(m=0, n+1, exp(m*x+x*O(x^n))*prod(k=1, m, exp(k*x+x*O(x^n))-1)), n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 16 2012
STATUS
approved