A207651 G.f.: Sum_{n>=0} Product_{k=1..n} (1 - (1-x)^k)/(1 - x^k).

1, 1, 3, 8, 25, 83, 323, 1410, 7062, 39660, 248287, 1709505, 12843315, 104446836, 913968191, 8560027375, 85427505885, 904899664970, 10139054456975, 119802780498730, 1488769376468607, 19409525611304801, 264890181139521141, 3776619220990535910

Offset: 0

Author

Paul D. Hanna, Feb 19 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 323*x^6 +...
such that, by definition,
A(x) = 1 + (1-(1-x))/(1-x) + (1-(1-x))*(1-(1-x)^2)/((1-x)*(1-x^2)) + (1-(1-x))*(1-(1-x)^2)*(1-(1-x)^3)/((1-x)*(1-x^2)*(1-x^3)) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..310
• Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Crossrefs

Cf. A207652, A207653, A207654, A193548.

Programs

PARI

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1-(1-x)^k)/(1-x^k +x*O(x^n)) )),n)}
for(n=0,25,print1(a(n),", "))

Formula

a(n) ~ 2*exp(Pi^2/12) * 6^(n+3/2) * n^(n+1) / (exp(n) * Pi^(2*n+2)). - Vaclav Kotesovec, Oct 31 2014