A207214 E.g.f.: Sum_{n>=0} exp(n*x) * Product_{k=1..n} (exp(k*x) - 1).

1, 1, 7, 85, 1759, 55621, 2501407, 151984645, 12004046719, 1196068161541, 146792747463007, 21762540250822405, 3834791755438306879, 792270319634586707461, 189687840256042278859807, 52103089179906338874671365, 16275196750916467736633834239

Offset: 0

Author

Paul D. Hanna, Feb 16 2012

Keywords

• nonn

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)).

Examples

			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.
		

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.

Crossrefs

Cf. A158690.

Programs

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),", "))

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