A207653 G.f.: Sum_{n>=0} Product_{k=1..n} (1 - (1-x)^(2*k-1))/(1 - x^(2*k-1)).

1, 1, 4, 16, 77, 460, 3287, 27561, 265307, 2880875, 34821316, 463543454, 6737545832, 106158368798, 1802204594518, 32793160634292, 636683459975767, 13137118248246982, 287070448575006268, 6622644707103106925, 160846900060253917905, 4102379491083664461080

Offset: 0

Author

Paul D. Hanna, Feb 19 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 77*x^4 + 460*x^5 + 3287*x^6 +...
such that, by definition,
A(x) = 1 + (1-(1-x))/(1-x) + (1-(1-x))*(1-(1-x)^3)/((1-x)*(1-x^3)) + (1-(1-x))*(1-(1-x)^3)*(1-(1-x)^5)/((1-x)*(1-x^3)*(1-x^5)) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200
• Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Crossrefs

Cf. A207651, A207652, A207654.

Programs

PARI

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1-(1-x)^(2*k-1))/(1-x^(2*k-1) +x*O(x^n)) )),n)}
for(n=0,40,print1(a(n),", "))

Formula

From Vaclav Kotesovec, Oct 31 2014: (Start)

a(n) ~ 6*sqrt(2) * exp(Pi^2/24) * 12^n * n! / Pi^(2*n+2).

a(n) ~ exp(Pi^2/24) * 12^(n+1) * n^(n+1/2) / (exp(n) * Pi^(2*n+3/2)).

(End)