A207556 G.f.: Sum_{n>=0} (1+x)^n * Product_{k=1..n} ((1+x)^k - 1).

1, 1, 3, 11, 55, 339, 2499, 21433, 209717, 2305719, 28141925, 377579731, 5523750291, 87508680045, 1492510215135, 27266981038343, 531245913925837, 10995334516297279, 240925208376757203, 5571653169126500083, 135617881389268715939, 3465772763274106884733

Offset: 0

Author

Paul D. Hanna, Feb 18 2012

Keywords

• nonn

Comments

Compare g.f. to: Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1), which is the g.f. of A179525.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 55*x^4 + 339*x^5 + 2499*x^6 +...
such that, by definition,
A(x) = 1 + (1+x)*((1+x)-1) + (1+x)^2*((1+x)-1)*((1+x)^2-1) + (1+x)^3*((1+x)-1)*((1+x)^2-1)*((1+x)^3-1) + (1+x)^4*((1+x)-1)*((1+x)^2-1)*((1+x)^3-1)*((1+x)^4-1) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..170
• Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Crossrefs

Cf. A179525, A207557.

Programs

PARI

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

Formula

a(n) ~ 2 * 12^(n+1) * n^(n+1/2) / (exp(n+Pi^2/24) * Pi^(2*n+3/2)). - Vaclav Kotesovec, May 07 2014