A214687 E.g.f.: Sum_{n>=0} exp(2*n*x) * Product_{k=1..n} (exp((2*k-1)*x) - 1).

1, 1, 11, 217, 7691, 430921, 35117531, 3927676537, 577640740331, 108115035641641, 25097054302205051, 7076531411753120857, 2382432541064412524171, 943997056642739165681161, 434864796716131476530668571, 230460477665217932140097413177

Offset: 0

Author

Paul D. Hanna, Aug 01 2012

Keywords

• nonn

Comments

Compare the e.g.f. to the identity:

exp(-x) = Sum_{n>=0} exp(2*n*x) * Product_{k=1..n} (1 - exp((2*k-1)*x)).

Examples

			E.g.f.: A(x) = 1 + x + 11*x^2/2! + 217*x^3/3! + 7691*x^4/4! + 430921*x^5/5! +...
such that, by definition,
A(x) = 1 + exp(2*x)*(exp(x)-1) + exp(4*x)*(exp(x)-1)*(exp(3*x)-1)
+ exp(6*x)*(exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)
+ exp(8*x)*(exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)*(exp(7*x)-1) +...
Compare this series to the identity:
exp(-x) = 1 - exp(2*x)*(exp(x)-1) + exp(4*x)*(exp(x)-1)*(exp(3*x)-1)
- exp(6*x)*(exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)
+ exp(8*x)*(exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)*(exp(7*x)-1)  +-...
The related e.g.f. of A215066 equals the series:
G(x) = 1 + (exp(x)-1) + (exp(x)-1)*(exp(3*x)-1)
+ (exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)
+ (exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)*(exp(7*x)-1) +...
or, more explicitly,
G(x) = 1 + x + 7*x^2/2! + 127*x^3/3! + 4315*x^4/4! + 235831*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..175
• Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Crossrefs

Cf. A207214, A215066.

Programs

PARI

{a(n)=n!*polcoeff(sum(m=0,n+1,exp(2*m*x+x*O(x^n))*prod(k=1,m,exp((2*k-1)*x+x*O(x^n))-1)),n)}
for(n=0,26,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((2*k-1)*x) - 1) = e.g.f. of A215066.

a(n) ~ 2*sqrt(6) * 24^n * (n!)^2 / (sqrt(n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, May 05 2014