A207652 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1)/(1 - x^k).

1, 1, 3, 10, 45, 249, 1709, 13912, 131168, 1402706, 16757321, 221018769, 3188425939, 49925523804, 843121969923, 15272776193787, 295372123082865, 6073931908657770, 132329525329523223, 3044691799670213778, 73771773281455834427, 1877511491197391256001

Offset: 0

Author

Paul D. Hanna, Feb 19 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 45*x^4 + 249*x^5 + 1709*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)-1)/(1-x) + ((1+x)-1)*((1+x)^2-1)/((1-x)*(1-x^2)) + ((1+x)-1)*((1+x)^2-1)*((1+x)^3-1)/((1-x)*(1-x^2)*(1-x^3)) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..210
• Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Crossrefs

Cf. A207651, A207653, A207654.

Programs

PARI

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,((1+x)^k-1)/(1-x^k +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) * 12^n * n! / (exp(Pi^2/24) * Pi^(2*n+2)).

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

(End)